Power Reduction Formulas

=== Square of Sine === {{:Power Reduction Formulas/Sine Squared}} === Square of Cosine === {{:Power Reduction Formulas/Cosine Squared}} === Square of Tangent === {{:Power Reduction Formulas/Tangent Squared}} === Cube of Sine === {{:Power Reduction Formulas/Sine Cubed}} === Cube of Cosine === {{:Power Reduction Formulas/Cosine Cubed}} === Fourth Power of Sine === {{:Power Reduction Formulas/Sine to 4th}} === Fourth Power of Cosine === {{:Power Reduction Formulas/Cosine to 4th}} === Fifth Power of Sine === {{:Power Reduction Formulas/Sine to 5th}} === Fifth Power of Cosine === {{:Power Reduction Formulas/Cosine to 5th}}


Countable Space is Sigma-Compact

Let $T = \left({S, \tau}\right)$ be a topological space, where $S$ is a countable set. Then $T$ is $\sigma$-compact.


Closed Sets of Fortissimo Space

Let $T = \struct {S, \tau_p}$ be a Fortissimo space. Then $H \subseteq S$ is closed in $T$ {{iff}}: :$p \in H$ or :$H$ is countable or both.


Union of Topologies on Singleton or Doubleton is Topology

Let $S$ be a set containing either exactly one or exactly two elements. Let $\family {\tau_i}_{i \mathop \in I}$ be an arbitrary non-empty indexed set of topologies for a set $S$. Then $\tau := \displaystyle \bigcup_{i \mathop \in I} {\tau_i}$ is also a topology for $S$.


Meager Sets in Arens-Fort Space

Let $T = \struct {S, \tau}$ be the Arens-Fort space. Let $A \subseteq S$. Then $A$ is meager {{iff}} $A = \set {\tuple {0, 0} }$.


Right and Left Regular Representations in Topological Group are Homeomorphisms

Let $\left({G, \cdot, \tau}\right)$ be a topological group. Let $g \in G$ be an element of $G$. Then the left and right regular representations with respect to $g$: :$L_g: \left({G, \tau}\right) \to \left({G, \tau}\right)$ and: :$R_g: \left({G, \tau}\right) \to \left({G, \tau}\right)$ are homeomorphisms.


Center of Group is Subgroup

Let $G$ be a group. The center $\map Z G$ of $G$ is a subgroup of $G$.


Center of Group is Abelian Subgroup

The center $Z \left({G}\right)$ of any group $G$ is a subgroup of $G$ which is abelian.


Dihedral Group/Group Presentation

The dihedral group $D_n$ has the group presentation: :$D_n = \gen {\alpha, \beta: \alpha^n = \beta^2 = e, \beta \alpha \beta = \alpha^{−1} }$ That is, the dihedral group $D_n$ is generated by two elements $\alpha$ and $\beta$ such that: :$(1): \quad \alpha^n = e$ :$(2): \quad \beta^2 = e$ :$(3): \quad \beta \alpha = \alpha^{n - 1} \beta$


Intersection with Subgroup Product of Superset

Let $X, Y, Z$ be subgroups of a group $\struct {G, \circ}$. Let $Y \subseteq X$. Then: :$X \cap \paren {Y \circ Z} = Y \circ \paren {X \cap Z}$ where $Y \circ Z$ denotes subset product.


Properties of Semi-Inner Product

Let $V$ be a vector space over $\Bbb F \in \left\{{\R, \C}\right\}$. Let $\left \langle{\cdot, \cdot}\right \rangle$ be a semi-inner product on $V$. Denote, for $x \in V$, $\left\Vert{x}\right\Vert := \left\langle{x, x}\right\rangle^{1 / 2}$. Then, $\forall x, y \in V, a \in \Bbb F$: :$(1): \quad \left\Vert{x + y}\right\Vert \le \left\Vert{x}\right\Vert + \left\Vert{y}\right\Vert$ :$(2): \quad \left\Vert{a x}\right\Vert = \left|{a}\right| \left\Vert{x}\right\Vert$


Inner Product Norm is Norm

Let $V$ be an inner product space over a subfield $\Bbb F$ of $\C$. Let $\norm {\, \cdot \,}$ denote the inner product norm on $V$. Then $\norm {\, \cdot \,}$ is a norm on $V$.


Metric Induced by Norm is Metric

Let $V$ be a normed vector space. Let $\norm{\,\cdot\,}$ denote its norm. Let $d$ be the metric induced by $\norm{\,\cdot\,}$. Then $d$ is a metric.


Product of Sums of Four Squares/Corollary

Let $a_1, a_2, \ldots, a_n, b_1, b_2, \ldots, b_n, c_1, c_2, \ldots, c_n, d_1, d_2, \ldots, d_n$ be integers. Then: : $\displaystyle \exists w, x, y, z \in \Z: \prod_{j \mathop = 1}^n \left({a_j^2 + b_j^2 + c_j^2 + d_j^2}\right) = w^2 + x^2 + y^2 + z^2$ That is, the product of any number of sums of four squares is also a sum of four squares.


Subgroup Subset of Subgroup Product

Let $\left({G, \circ}\right)$ be a group whose identity is $e$. Let $H$ and $K$ be subgroups of $G$. Then: :$H \subseteq H \circ K \supseteq K$ where $H \circ K$ denotes the subset product of $H$ and $K$.


Product Space is Path-connected iff Factor Spaces are Path-connected

Let $\SS = \family {\struct{S_i, \tau_i}}_{i \in I}$ be an indexed family of topological spaces for $i$ in some indexing set $I$ such that $\forall i \in I: S_i \ne \varnothing$. Let $\displaystyle T = \struct{S, \tau} = \prod_{i \mathop \in I} \struct{S_i, \tau_i}$ be the product space of $\SS$. Then $T$ is a path-connected space {{iff}} each of $\struct{S_i, \tau_i}$ is a path-connected space.


Normal Subgroup of Subset Product of Subgroups

Let $G$ be a group whose identity is $e$. Let: : $H$ be a subgroup of $G$ : $N$ be a normal subgroup of $G$. Then: : $N \lhd N H$ where: : $\lhd$ denotes normal subgroup : $N H$ denotes subset product.


Non-Abelian Order 8 Group with One Order 2 Element is Quaternion Group

Let $G$ be a group with the following properties: :$(1): \quad G$ is non-abelian. :$(2): \quad G$ is of order $8$. :$(3): \quad G$ has precisely one element of order $2$. Then $G$ is isomorphic to the quaternion group $Q$.


Subset Product of Normal Subgroups with Trivial Intersection

Let $\struct {G, \circ}$ be a group whose identity is $e$. Let $H, K$ be normal subgroups of $G$. Let $H \cap K = e$. Then $H K$ is isomorphic to $H \times K$ where: : $H K$ denotes the subset product of $H$ and $K$ : $H \times K$ denotes the direct product of $H$ and $K$.


Completion Theorem (Inner Product Space)

Let $V$ be an inner product space over a subfield $\Bbb F$ of $\C$. Let $\left\langle{\cdot, \cdot}\right\rangle_V$ be the inner product on $V$. Let $d: V \times V \to \R_{\ge 0}$ be the metric induced by the inner product norm. Let $H$ be the completion of $V$ with respect to $d$. Then $\left\langle{\cdot, \cdot}\right\rangle_V$ can be extended to an inner product on $H$. By definition, $H$ will be a Hilbert space. Therefore, the theorem can alternatively be stated as: :Any inner product space may be completed to a Hilbert space.


Identity Mapping is Group Endomorphism

Let $\struct {G, \circ}$ be a group whose identity is $e$. Then $I_G: \struct {G, \circ} \to \struct {G, \circ}$ is a group endomorphism.


Pythagoras's Theorem (Hilbert Space)

Let $H$ be a Hilbert space with inner product norm $\norm {\, \cdot \,}$. Let $f_1, \ldots, f_n \in H$ be pairwise orthogonal. Then: :$\displaystyle \norm {\sum_{i \mathop = 1}^n f_i}^2 = \sum_{i \mathop = 1}^n \norm {f_i}^2$


Primitive of Reciprocal of x squared plus a squared/Arctangent Form

:$\ds \int \frac {\d x} {x^2 + a^2} = \frac 1 a \arctan \frac x a + C$


Endomorphism from Integers to Multiples

Let $\struct {\Z, +}$ be the additive group of integers. Let $\phi: \struct {\Z, +} \to \struct {\Z, +}$ be a mapping. Then $\phi$ is a group endomorphism {{iff}}: :$\exists k \in \Z: \forall n \in \Z: \map \phi n = k n$


Finite Cyclic Group is Isomorphic to Integers under Modulo Addition

Let $\struct {G, \circ}$ be a finite group whose identity element is $e$. Then $\struct {G, \circ}$ is cyclic of order $n$ {{iff}} $\struct {G, \circ}$ is isomorphic with the additive group of integers modulo $n$ $\struct {\Z_n, +_n}$.


Parallelogram Law (Hilbert Space)

Let $H$ be a Hilbert space with associated norm $\norm {\, \cdot \,}$. Let $f, g \in H$ be arbitrary. Then: :$\norm {f + g}^2 + \norm {f - g}^2 = 2 \paren {\norm f^2 + \norm g^2}$


Unique Point of Minimal Distance

Let $H$ be a Hilbert space, and let $h \in H$. Let $K \subseteq H$ be a closed, convex, non-empty subset of $H$. Then there is a unique point $k_0 \in K$ such that: :$\norm {h - k_0} = \map d {h, K}$ where $d$ denotes distance to a set. {{refactor|Own page, and prove it}} Furthermore, if $K$ is a linear subspace, this point is characterised by: :$\norm {h - k_0} = \map d {h, K} \iff \paren {h - k_0} \perp K$ where $\perp$ signifies orthogonality.


Primitive of Reciprocal of Root of a squared minus x squared/Arcsine Form

:$\displaystyle \int \frac 1 {\sqrt {a^2 - x^2} } \rd x = \arcsin \frac x a + C$


Inner Automorphism is Automorphism

Let $G$ be a group. Let $x \in G$. Let $\kappa_x$ be the inner automorphism of $x$ in $G$. Then $\kappa_x$ is an automorphism of $G$.


Order of Automorphism Group

Let $G$ be a finite group whose order is greater than $2$. Let $\Aut G$ be the automorphism group of $G$. Then the order of $\Aut G$ is greater than $1$.


Properties of Orthogonal Projection

Let $H$ be a Hilbert space. Let $K$ be a closed linear subspace of $H$. Let $P_K$ denote the orthogonal projection on $K$. Then $P_K$ has the following properties: :$(1):\qquad P_K$ is a linear transformation on $H$. :$(2):\qquad \forall h\in H: \left\|{P_K(h)}\right\| \le \left\|{h}\right\|$ :$(3):\qquad P_K \circ P_K = P_K$ :$(4):\qquad \ker P_K = K^{\perp}$ and $\operatorname{ran} P_K = K$ {{wtd|I am not entirely happy with this setup of the page, with four results stated. Strictly speaking, each should have its own page. But I am not up to thinking of a name for each of the results separately. Probably I will put up four transcluded proof pages. Lastly, note that these four results are one theorem $I.2.7$ in Conway. --LF

Four transcluded subpages is a technique that's been used before for this sort of linked set of results. The fact they're all part of the same proof in Conway is incidental. -- pm}}


Quotient Theorem for Group Homomorphisms

Let $\phi: G \to G'$ be a (group) homomorphism between two groups $G$ and $G'$. Then $\phi$ can be decomposed into the form: :$\phi = \alpha \beta \gamma$ where: :$\alpha: \Img \phi \to G'$ is a monomorphism :$\beta: G / \map \ker \phi \to \Img \phi$ is an isomorphism :$\gamma: G \to G / \map \ker \phi$ is an epimorphism.


Inclusion Mapping is Monomorphism

Let $\left({S, \circ}\right)$ be an algebraic structure. Let $\left({T, \circ}\right)$ be an algebraic substructure of $S$. Let $\iota: T \to S$ be the inclusion mapping from $T$ to $S$. Then $\iota$ is a monomorphism.


Schreier-Zassenhaus Theorem

Let $G$ be a finite group. Let $\HH_1$ and $\HH_2$ be two normal series for $G$. Then $\HH_1$ and $\HH_2$ have refinements of equal length whose factors are isomorphic.


Jordan-Hölder Theorem

Let $G$ be a finite group. Let $\HH_1$ and $\HH_2$ be two composition series for $G$. Then: :$\HH_1$ and $\HH_2$ have the same length :Corresponding factors of $\HH_1$ and $\HH_2$ are isomorphic.


Double Orthocomplement is Closed Linear Span

Let $\HH$ be a Hilbert space. Let $A \subseteq \HH$ be a subset of $\HH$. Then the following identity holds: :$\paren {A^\perp}^\perp = \vee A$ Here $A^\perp$ denotes orthocomplementation, and $\vee A$ denotes the closed linear span.


Orthocomplement is Closed Linear Subspace

Let $H$ be a Hilbert space. Let $A \subseteq H$ be a subset of $H$. Then the orthocomplement $A^\perp$ of $A$ is a closed linear subspace of $H$.


Linear Subspace Dense iff Zero Orthocomplement

Let $H$ be a Hilbert space. Let $K$ be a linear subspace of $H$. Then $K$ is everywhere dense {{iff}} $K^\perp = \left({0}\right)$, where $K^\perp$ is the orthocomplement of $K$, and $\left({0}\right)$ denotes the zero subspace.


Finite Group has Composition Series

Let $G$ be a finite group. Then $G$ has a composition series.


Length of Subgroup Plus Length of Quotient Group

Let $G$ be a finite group. Let $H$ be a normal subgroup of $G$. Then: :$\map l G = \map l H + \map l {G / H}$ where: :$\map l G$ denotes the length of $G$ :$G / H$ denotes the quotient group of $G$ by $H$.


Length of Abelian Group

Let $G$ be an abelian group whose order is $n$. Let $n$ have the prime decomposition: :$\displaystyle n = \prod_{i \mathop = 1}^r p_i^{k_i} = p_1^{k_1} p_2^{k_2} \cdots p_r^{k_r}$ where $p_1 < p_2 < \cdots < p_r$ are distinct primes and $k_1, k_2, \ldots, k_r$ are positive integers. Then the length of $G$ is given by: :$\displaystyle \map l G = \sum_{i \mathop = 1}^r k_i = k_1 + k_2 + \cdots + k_r$


Condition for Composition Series

Let $G$ be a finite group. Then: :a normal series $\HH$ for $G$ is a composition series for $G$ {{iff}}: :every factor group of $\HH$ is a simple group.


Factors of Composition Series for Prime Power Group

Let $G$ be a group such that $\order G = p^n$ where $p$ is a prime number. Then $G$ has a composition series in which each factor group is cyclic of order $p$.


Continuity of Linear Functionals

Let $H$ be a Hilbert space, and let $L$ be a linear functional on $H$. Then the following four statements are equivalent: :$(1):\quad L$ is continuous :$(2):\quad L$ is continuous at $\mathbf{0}_H$ :$(3):\quad L$ is continuous at some point :$(4):\quad \exists c > 0: \forall h \in H: \left|{Lh}\right| \le c \left\|{h}\right\|$


Equivalence of Definitions of Closed Linear Span

Let $H$ be a Hilbert space over $\Bbb F \in \set {\R, \C}$, and let $A \subseteq H$ be a subset. {{TFAE|def = Closed Linear Span|view = closed linear span of $A$}} :$(1): \quad \displaystyle \vee A = \bigcap \Bbb M$, where $\Bbb M$ consists of all closed linear subspaces $M$ of $H$ with $A \subseteq M$ :$(2): \quad \displaystyle \vee A$ is the smallest closed linear subspace $M$ of $H$ with $A \subseteq M$ :$(3): \quad \displaystyle \vee A = \map \cl {\set {\sum_{k \mathop = 1}^n \alpha_k f_k: n \in \N_{\ge 1}, \alpha_i \in \Bbb F, f_i \in A} }$, where $\cl$ denotes closure


Equivalence of Definitions of Norm of Linear Functional

Let $H$ be a Hilbert space, and let $L$ be a bounded linear functional on $H$. {{explain|We can weaken the assumption that $H$ be a Hilbert space by saying that it is just a normed space, I think. It is not strictly necessary to consider a Hilbert space.}} Define the following norms of $L$: :$(1): \quad \left\|{L}\right\|_1 = \sup \left\{{\left|{Lh}\right|: \left\|{h}\right\| \le 1}\right\}$ :$(2): \quad \left\|{L}\right\|_2 = \sup \left\{{\left|{Lh}\right|: \left\|{h}\right\| = 1}\right\}$ :$(3): \quad \left\|{L}\right\|_3 = \displaystyle \sup \left\{{\dfrac {\left|{Lh}\right|} {\left\|{h}\right\|}: h \in H\setminus \left\{\mathbf 0 \right\}}\right\}$ :$(4): \quad \left\|{L}\right\|_4 = \inf \left\{{c > 0: \forall h \in H: \left|{Lh}\right| \le c \left\|{h}\right\|}\right\}$ Then: : $\left\|{L}\right\|_1 = \left\|{L}\right\|_2 = \left\|{L}\right\|_3 = \left\|{L}\right\|_4$


Riesz Representation Theorem (Hilbert Spaces)

Let $H$ be a Hilbert space. Let $L$ be a bounded linear functional on $H$. Then there is a unique $h_0 \in H$ such that: :$\forall h \in H: L h = \innerprod h {h_0}$


Orthonormal Subset Extends to Basis

Let $H$ be a Hilbert space, and let $S$ be an orthonormal subset of $H$. Then there exists a basis for $H$ that contains $S$ as a subset.


Gram-Schmidt Orthogonalization

Let $H$ be a Hilbert space. Let $S = \set {h_n: n \in \N}$ be a linearly independent subset of $H$.  Then there exists an orthonormal subset $E = \set {e_n: n \in \N}$ of $H$ such that: :$\forall k \in \N: \operatorname{span} \set {h_n: 0 \le n \le k} = \operatorname{span} \set {e_n: 0 \le n \le k}$ where $\operatorname{span}$ denotes linear span.


Orthogonal Projection onto Closed Linear Span

Let $H$ be a Hilbert space, and let $E = \left\{{e_1, \ldots, e_n}\right\}$ be an orthonormal subset of $H$. Let $M = \vee E$, the closed linear span of $E$.  Then the orthogonal projection $P$ onto $M$ satisfies, $\forall h \in H$: :$Ph = \displaystyle \sum_{k=1}^n \left\langle{h, e_k}\right\rangle e_k$


Bessel's Inequality

Let $H$ be a Hilbert space, and let $E = \left\{{e_n: n \in \N}\right\}$ be a countably infinite orthonormal subset of $H$. Then, for all $h \in H$, one has the inequality: :$\displaystyle \sum_{n \mathop = 1}^\infty \left|{\left\langle{h, e_n}\right\rangle}\right|^2 \le \left\|{h}\right\|^2$


Between two Rational Numbers exists Irrational Number

Let $a, b \in \Q$ where $a < b$.  Then: :$\exists \xi \in \R \setminus \Q: a < \xi < b$


Abelian Group is Simple iff Prime

Let $G$ be a non-trivial abelian group. Then $G$ is simple {{iff}} $G$ is a prime group.


Cyclic Group is Simple iff Prime

Let $G$ be a cyclic group. Then $G$ is simple {{iff}} $G$ is a prime group.


Factors of Solvable Group are Prime

Let $G$ be a solvable group. Let $\mathcal H$ be a composition series of $G$. Then all factor groups of $\mathcal H$ must be prime.


Singleton is Convex Set

Let $V$ be a vector space over $\R$ or $\C$, and let $v \in V$. Then the singleton $S = \left\{{v}\right\}$ is a convex set.


Group is Solvable iff Normal Subgroup and Quotient are Solvable

Let $G$ be a finite group. Let $H$ be a normal subgroup of $G$. Then $G$ is solvable {{iff}}: :$(1): \quad H$ is solvable and :$(2): \quad G / H$ is solvable where $G / H$ is the quotient group of $G$ by $H$.


Intersection of Convex Sets is Convex Set (Vector Spaces)

Let $V$ be a vector space over $\R$ or $\C$. Let $\mathcal C$ be a family of convex subsets of $V$. Then the intersection $\displaystyle \bigcap \mathcal C$ is also a convex subset of $V$.


Prime Power Group is Solvable

Let $G$ be a group whose order is $p^n$ where $p$ is a prime number and $n$ is a positive integer. Then $G$ is solvable.


Subgroup of Solvable Group is Solvable

Let $G$ be a solvable group. Let $H$ be a subgroup of $G$. Then $H$ is solvable.


Group with Normal Series with Solvable Factor Groups is Solvable

Let $G$ be a solvable group. Let $\HH$ be a normal series for $G$. Let all the factor groups of $\HH$ be solvable. Then $G$ is solvable.


Group with Order Less than 60 is Solvable

Every group whose order is less than $60$ is solvable.


Direct Product of Solvable Groups is Solvable

Let $G$ and $H$ be groups which are solvable. Then their (external) direct product $G \times H$ is also solvable.


Group with One Sylow Subgroup per Prime Divisor is Solvable

Let $G$ be a group of order $n$. Suppose that, for each prime number $p$ which divides $n$, $G$ has exactly one $p$-Sylow subgroup. Then $G$ is solvable.


Permutation Group as Effective Transformation Group

Let $*$ be a group action of $G$ on $X$. Then $G$ is a permutation group {{iff}} $G$ acts acts effectively on $X$.


Characterization of Bases (Hilbert Spaces)

Let $H$ be a Hilbert space, and let $E$ be an orthonormal subset of $H$. Then the following six statements are equivalent: :$(1): \qquad E$ is a basis for $H$ :$(2): \qquad h \in H, h \perp E \implies h = \mathbf 0$, where $\perp$ denotes orthogonality :$(3): \qquad \vee E = H$, where $\vee E$ denotes the closed linear span of $E$ :$(4): \qquad \forall h \in H: h = \displaystyle \sum \left\{{\left\langle{h, e}\right\rangle e: e \in E}\right\}$ :$(5): \qquad \forall g, h \in H: \left\langle{g, h}\right\rangle = \displaystyle \sum \left\{{\left\langle{g, e}\right\rangle\left\langle{e, h}\right\rangle: e \in E}\right\}$ :$(6): \qquad \forall h \in H: \left\Vert{h}\right\Vert^2 = \displaystyle \sum \left\{{\left\vert{\left\langle{h, e}\right\rangle}\right\vert^2: e \in E}\right\}$ In the last three statements, $\displaystyle \sum$ denotes a generalized sum. Statement $(6)$ is commonly known as '''Parseval's identity'''.


Square Number Less than One

Let $x$ be a real number such that $x^2 < 1$. Then: : $x \in \left({-1 \,.\,.\, 1}\right)$ where $\left({-1 \,.\,.\, 1}\right)$ is the open interval $\left\{{x \in \R: -1 < x < 1}\right\}$.


Subgroups of Symmetric Group Isomorphic to Product of Subgroups

Let $S_n$ denote the symmetric group on $n$ letters. Let $k \in \closedint 1 n$. Then there are $\dbinom n k$ subgroups of $S_n$ which are isomorphic to $S_k \times S_{n - k}$, where $\dbinom n k$ denotes the binomial coefficient. All of these $\dbinom n k$ subgroups are conjugate.


Uniform Limit Theorem

Let $\left({M, d_M}\right)$ and $\left({N, d_N}\right)$ be metric spaces. Let $\left \langle{f_n}\right \rangle$ be a sequence of mappings from $M$ to $N$ such that: :$(1): \quad \forall n \in \N: f_n$ is continuous at every point of $M$ :$(2): \quad  \langle f_n\rangle$ converges uniformly to $f$ Then: : $f$ is continuous at every point of $M$.


Equality is Transitive

:$\forall a, b, c: \paren {a = b} \land \paren {b = c} \implies a = c$


Axiom:Axioms of Equality

The '''axioms of equality''' are strictly speaking not axiomatic at all, as they can be deduced from still more basic axioms, in particular Leibniz's law: :$x = y \dashv \vdash P \left({x}\right) \iff P \left({y}\right)$ where $P \left({x}\right)$ and $P \left({y}\right)$ are propositional functions on the elements $x$ and $y$ of the universe of discourse.


Axiom of Pairing from Infinity and Replacement

The Axiom of Pairing is a consequence of: :the Axiom of Infinity and :the Axiom of Replacement.


Axiom of Pairing from Powers and Replacement

The Axiom of Pairing is a consequence of: :the Axiom of the Empty Set and :the Axiom of Powers and also :the Axiom of Replacement.


Axiom of Pairing from Axiom of Subsets

Let it be supposed that there exists a set which contains at least two elements. Then the Axiom of Pairing is a consequence of the Axiom of Specification.


Existence of Singleton Set

Let $a$ be a set. Then the singleton set $\set a$ may be constructed such that: :$a \in \set a$


Intersection of Unions with Complements is Subset of Union

Let $R, S, T$ be sets. Then: :$\left({R \cup S}\right) \cap \left({\overline R \cup T}\right) \subseteq S \cup T$


Conjunction implies Disjunction of Conjunctions with Complements

:$p \land q \vdash \left({p \land r}\right) \lor \left({q \land \neg r}\right)$


Conjunction of Disjunctions with Complements implies Disjunction

:$\paren {p \lor r} \land \paren {q \lor \neg r} \vdash p \lor q$


Intersection of Elements of Power Set

Let $S$ be a set. Let: : $\displaystyle \mathbb S = \bigcap_{X \mathop \in \mathcal P \left({S}\right)} X$ where $\mathcal P \left({S}\right)$ is the power set of $S$. Then $\mathbb S = \varnothing$.


Power Set of Subset

Let $S \subseteq T$ where $S$ and $T$ are both sets. Then: :$\powerset S \subseteq \powerset T$ where $\powerset S$ denotes the power set of $S$.


Equivalence of Definitions of Ordered Pair

The following definitions of an ordered pair are equivalent:


Subset of Cartesian Product

Let $S$ be a set of ordered pairs. Then $S$ is the subset of the cartesian product of two sets.


Continuously Differentiable Curve has Finite Arc Length

Let $y = \map f x$ be a real function which is continuous on the closed interval $\closedint a b$ and continuously differentiable on the open interval $\openint a b$. The definite integral:  :$s = \displaystyle \int_{x \mathop = a}^{x \mathop = b} \sqrt {1 + \paren {\frac {\d y} {\d x} }^2} \rd x$ exists, and is called the '''arc length''' of $f$ between $a$ and $b$.


Arc Length for Parametric Equations

Let $x = \map f t$ and $y = \map g t$ be real functions of a parameter $t$.  Let these equations describe a curve $\mathcal C$ that is continuous for all $t \in \closedint a b$ and continuously differentiable for all $t \in \openint a b$. Suppose that the graph of the curve does not intersect itself for any $t \in \openint a b$. Then the arc length of $\mathcal C$ between $a$ and $b$ is given by: :$s = \displaystyle \int_a^b \sqrt {\paren {\frac {\d x} {\d t} }^2 + \paren {\frac {\d y} {\d t} }^2} \rd t$ for $\dfrac {\d x} {\d t} \ne 0$.


Inverse Element of Injection

Let $S$ and $T$ be sets. Let $f: S \to T$ be an injection. Then: :$\map {f^{-1} } y = x \iff \map f x = y$


Hilbert Space Separable iff Countable Dimension

Let $H$ be a Hilbert space. Then $H$ is a separable space iff it has countable dimension.


Net Convergence Equivalent to Absolute Convergence

Let $V$ be a Banach space. Let $\left({v_n}\right)_{n \in \N}$ be a sequence of elements in $V$. Then the following two statements are equivalent: :$(1): \qquad \displaystyle \sum_{n=1}^\infty \left\Vert{v_n}\right\Vert$ converges (absolute convergence) :$(2): \qquad \displaystyle \sum \left\{{v_n: n \in \N}\right\}$ converges (generalised or net convergence)


Intersection of Infinite Successor Sets

Let $\mathbb S$ be a non-empty indexed family of infinite successor sets. Then $\bigcap \mathbb S$ is an infinite successor set.


Minimal Infinite Successor Set Exists

There exists a minimal infinite successor set $\omega$ that is a subset of every other infinite successor set.


Minimal Infinite Successor Set forms Peano Structure

Let $\omega$ be the minimal infinite successor set. Let $\cdot^+: \omega \to \omega$ be the mapping assigning to a set its successor set: :$n^+ := n \cup \left\{{n}\right\}$ Let $\varnothing \in \omega$ be the empty set. Then $\left({\omega, \cdot^+, \varnothing}\right)$ is a Peano structure.


Integral of Logarithm

:$\displaystyle \int \ln x \ \mathrm d x = x \ln x - x + C$ :$\displaystyle \int \log_a x \ \mathrm d x =  \frac 1 {\ln a}\left({x \ln x - x}\right) + C$ for $x > 0$.


Finite Ordinal is not Subset of one of its Elements

Let $n$ be a finite ordinal. Then: :$\nexists x \in n: n \subseteq x$ that is, $n$ is not a subset of one of its elements.


Element of Minimal Infinite Successor Set is Transitive Set

Let $\omega$ be the minimal infinite successor set. Let $n \in \omega$. Then $x \in n \implies x \subseteq n$. That is, every element of $n$ is also a subset of it. In other words, each element of $\omega$ is a transitive set.


Primitive of Exponential Function

:$\displaystyle \int e^x \rd x = e^x + C$ where $C$ is an arbitrary constant.


Natural Number Addition Commutes with Zero

Let $\N$ be the natural numbers. Then: :$\forall n \in \N: 0 + n = n = n + 0$


Natural Number Addition Commutativity with Successor

Let $\N$ be the natural numbers. Then: :$\forall m, n \in \N: m^+ + n = \paren {m + n}^+$


Natural Numbers are Comparable

Let $\N$ be the natural numbers. Let $m, n \in \N$. Then $m$ and $n$ are comparable by the ordering relation $\le$. That is, either: :$(1): \quad m \le n$ or: :$(2): \quad n \le m$ or possibly both.


Zero is Zero Element for Natural Number Multiplication

Let $\N$ be the natural numbers. Then $0$ is a zero element for multiplication: :$\forall n \in \N: 0 \times n = 0 = n \times 0$


Hilbert Space Isomorphism is Equivalence Relation

Hilbert space isomorphism is an equivalence relation.


Element of Finite Ordinal iff Subset

Let $m, n$ be distinct finite ordinals. Then: :$m \in n \iff m \subseteq n$


Ordering on Natural Numbers is Compatible with Addition

Let $m, n, k \in \N$ where $\N$ is the set of natural numbers. Then: :$m < n \iff m + k < n + k$


Hilbert Space Isomorphism is Bijection

Let $H, K$ be Hilbert spaces. Denote by $\left\langle{\cdot, \cdot}\right\rangle_H$ and $\left\langle{\cdot, \cdot}\right\rangle_K$ their respective inner products. Let $U: H \to K$ be an isomorphism. Then $U$ is a bijection.


Hilbert Spaces Isomorphic iff Same Dimension

Two Hilbert spaces are isomorphic iff they have the same dimension.


Surjection that Preserves Inner Product is Linear

Let $H, K$ be Hilbert spaces, and denote by $\left\langle{\cdot, \cdot}\right\rangle_H$ and $\left\langle{\cdot, \cdot}\right\rangle_K$ their respective inner products. Let $U: H \to K$ be a surjection such that: :$\forall g,h \in H: \left\langle{g, h}\right\rangle_H = \left\langle{Ug, Uh}\right\rangle_K$ Then $U$ is a linear map, and hence an isomorphism.


Linear Operator on General Logarithm

Let $\phi: \R^\R \to \R^\R, y \mapsto \phi\left({y}\right)$ be a linear operator on the space of functions from $\R\to\R$. Let $y$ be a real function such that $\forall x \in \R$, $y\left({x}\right) > \mathbf{0}\left({x}\right) = 0$. Let $\log_a y$ be the logarithm of $y$ to base $a$. Then: :$\phi \left({\log_a y}\right) = \dfrac 1 {\ln a} \left({\phi \left({\ln y}\right)}\right)$ where $\ln$ is the natural logarithm.


Ordering on Natural Numbers is Compatible with Multiplication

Let $m, n, k \in \N$ where $\N$ is the set of natural numbers. Let $k \ne 0$. Then: :$m < n \iff m \times k < n \times k$


Natural Number Ordering is Transitive

Let $m, n, k \in \N$ where $\N$ is the set of natural numbers. Let $<$ be the relation defined on $\N$ such that: :$m < n \iff m \in n$ where $\N$ is defined as the minimal infinite successor set $\omega$. Then: :$k < m, m < n \implies k < n$ That is: $<$ is a transitive relation.


Proper Subset of Finite Ordinal is Equivalent to Smaller Ordinal

Let $n$ be a finite ordinal. Let $x \subsetneq n$. Then for some finite ordinal $m < n$: :$m \sim x$ where $m \sim x$ denotes that $m$ is (set) equivalent to $x$. That is, every proper subset of a finite ordinal $n$ is equivalent to some finite ordinal smaller than $n$.


Hilbert Space Direct Sum is Hilbert Space

Let $\left({H_i}\right)_{i \in I}$ be a $I$-indexed family of Hilbert spaces over $\Bbb F \in \left\{{\R, \C}\right\}$. Let $H = \displaystyle \bigoplus_{i \mathop \in I} H_i$ be their Hilbert space direct sum. Then $H$ is a Hilbert space.


Morera's Theorem

Let $D$ be a simply connected domain in $\C$. Let $f: D \to \C$ be a continuous function. Let $f$ be such that: :$\displaystyle \int_\gamma \map f z \rd z = 0$ for every simple closed contour $\gamma$ in $D$ Then $f$ is analytic on $D$.


Relation is Antisymmetric iff Intersection with Inverse is Coreflexive

Let $\RR$ be a relation on $S$. Then: :$\RR$ is antisymmetric {{iff}}: :$\RR \cap \RR^{-1}$ is coreflexive where $\RR^{-1}$ is the inverse of $\RR$.


Min Operation on Toset is Semigroup

Let $\left({S, \preceq}\right)$ be a totally ordered set. Let $\min \left({x, y}\right)$ denote the min operation on $x, y \in S$. Then $\left({S, \min}\right)$ is a semigroup.


Max Semigroup is Commutative

Let $\left({S, \preceq}\right)$ be a totally ordered set.  Then the semigroup $\left({S, \max}\right)$ is commutative.


Relation is Connected iff Union with Inverse and Diagonal is Trivial Relation

Let $\mathcal R$ be a relation on $S$. Then $\mathcal R$ is a connected relation {{iff}}: :$\mathcal R \cup \mathcal R^{-1} \cup \Delta_S = S \times S$ where $\mathcal R^{-1}$ is the inverse of $\mathcal R$ and $\Delta_S$ is the diagonal relation.


Absolutely Convergent Generalized Sum Converges

Let $V$ be a Banach space. Let $\norm {\, \cdot \,}$ denote the norm on $V$. Let $d$ denote the corresponding induced metric. Let $\family {v_i}_{i \mathop \in I}$ be an indexed subset of $V$ such that the generalized sum $\displaystyle \sum_{i \mathop \in I} \set {v_i}$ converges absolutely. Then the generalized sum $\displaystyle \sum \set {v_i: i \in I}$ converges.


Generalized Sum is Monotone

Let $\left({a_i}\right)_{i \in I}$ be an $I$-indexed family of positive real numbers. That is, let $a_i \in \R_{\ge 0}$ for all $i \in I$. Then, for every finite subset $F$ of $I$: :$\displaystyle \sum_{i \mathop \in F} a_i \le \sum \left\{{a_i : i \in I}\right\}$ provided the generalized sum on the right converges.


Convergent Generalized Sum of Positive Reals has Countably Many Non-Zero Terms

Let $\left({a_i}\right)_{i \mathop \in I}$ be an $I$-indexed family of positive real numbers. That is, let $a_i \in \R_{\ge 0}$ for all $i \in I$. Suppose that $\displaystyle \sum \left\{ {a_i: i \in I}\right\}$ converges. Then the set $I_{>0} := \left\{ {i \in I: a_i > 0}\right\}$ is countable.


Finite Subsets form Directed Set

Let $I$ be a set. Denote with $\mathcal F$ the set of finite subsets of $I$. Let $\subseteq$ be the subset relation on $\mathcal F$. Then $\struct {\mathcal F, \subseteq}$ is a directed set.


Strictly Precedes is Strict Ordering

Let $\left({S, \preceq}\right)$ be an ordered set. Let $\prec$ be the relation on $S$ defined as: :$a \prec b \iff (a \ne b) \land (a \preceq b)$ That is, $a \prec b$ {{iff}} $a$ strictly precedes $b$. Then: : $a \preceq b \iff (a = b) \lor (a \prec b)$ and $\prec$ is a strict ordering on $S$.


Smallest Element is Unique

Let $\struct {S, \preceq}$ be an ordered set. If $S$ has a smallest element, then it can have only one. That is, if $a$ and $b$ are both smallest elements of $S$, then $a = b$.


Greatest Element is Unique

Let $\struct {S, \preceq}$ be a ordered set. If $S$ has a greatest element, then it can have only one. That is, if $a$ and $b$ are both greatest elements of $S$, then $a = b$.


Smallest Element is Lower Bound

Let $\left({S, \preceq}\right)$ be an ordered set. Let $T \subseteq S$. Let $T$ have a smallest element $m \in T$. Then $m$ is a lower bound of $T$. It follows by definition that $T$ is bounded below.


Greatest Element is Upper Bound

Let $\struct {S, \preceq}$ be an ordered set. Let $T \subseteq S$. Let $T$ have a greatest element $M \in T$. Then $M$ is an upper bound of $T$. It follows by definition that $T$ is bounded above.


Intersection of Subset with Lower Bounds

Let $\struct {S, \preceq}$ be an ordered set. Let $T \subseteq S$. Let $T_*$ be the set of all lower bounds of $T$ in $S$. Then $T_* \cap T \ne \O$ {{iff}}: :$T$ has a smallest element $m$ and :$T_* \cap T$ is a singleton such that $T_* \cap T = \set m$


Nesbitt's Inequality

Let $a$, $b$ and $c$ be positive real numbers. Then: :$\dfrac a {b + c} + \dfrac b {a + c} + \dfrac c {a + b} \ge \dfrac 3 2$


Greatest Element is Maximal

Let $\struct {S, \preceq}$ be an ordered set which has a greatest element. Let $M$ be the greatest element of $\struct {S, \preceq}$. Then $M$ is a maximal element.


Smallest Element is Minimal

Let $\struct {S, \preceq}$ be an ordered set which has a smallest element. Let $m$ be the smallest element of $\struct {S, \preceq}$. Then $m$ is a minimal element.


Minimal Element in Toset is Unique and Smallest

Let $\struct {S, \preceq}$ be a totally ordered set. Let $m$ be a minimal element of $\struct {S, \preceq}$. Then: :$(1): \quad m$ is the smallest element of $\struct {S, \preceq}$ :$(2): \quad m$ is the only minimal element of $\struct {S, \preceq}$.


Maximal Element in Toset is Unique and Greatest

Let $\struct {S, \preceq}$ be a totally ordered set. Let $M$ be a maximal element of $\struct {S, \preceq}$. Then: :$(1): \quad M$ is the greatest element of $\struct {S, \preceq}$. :$(2): \quad M$ is the only maximal element of $\struct {S, \preceq}$.


Whitney Embedding Theorem

Every smooth $m$-dimensional manifold admits a smooth embedding into Euclidean space $\R^{2m+1}$.


Max Semigroup is Idempotent

Let $\struct {S, \preceq}$ be a totally ordered set. Then the semigroup $\struct {S, \max}$ is an idempotent semigroup.


Tarski's Geometry is Complete

Tarski's geometry is both consistent and complete.


Continuity of Linear Transformations

Let $H, K$ be Hilbert spaces, and let $A: H \to K$ be a linear transformation. Then the following four statements are equivalent: :$(1): \quad A$ is continuous :$(2): \quad A$ is continuous at $\mathbf 0_H$ :$(3): \quad A$ is continuous at some point :$(4): \quad \exists c > 0: \forall h \in H: \norm {\map A h}_K \le c \norm h_H$


Equivalence of Definitions of Norm of Linear Transformation

Let $H, K$ be Hilbert spaces, and let $A: H \to K$ be a bounded linear transformation. {{TFAE|def = Norm on Bounded Linear Transformation}} :$(1): \qquad \left\Vert{A}\right\Vert = \sup \left\{{\left\Vert{Ah}\right\Vert_K: \left\Vert{h}\right\Vert_H \le 1}\right\}$ :$(2): \qquad \left\Vert{A}\right\Vert = \sup \left\{{\dfrac {\left\Vert{Ah}\right\Vert_K} {\left\Vert{h}\right\Vert_H}: h \in H, h \ne \mathbf{0}_H}\right\}$ :$(3): \qquad \left\Vert{A}\right\Vert = \sup \left\{{\left\Vert{Ah}\right\Vert_K: \left\Vert{h}\right\Vert_H \le 1}\right\}$ :$(4): \qquad \left\Vert{A}\right\Vert = \inf \left\{{c > 0: \forall h \in H: \left\Vert{Ah}\right\Vert_K \le c \left\Vert{h}\right\Vert_H}\right\}$


Max Semigroup on Toset is Semilattice

Let $\struct {S, \preceq}$ be a totally ordered set.  Then the max semigroup $\struct {S, \max}$ is a semilattice.


Max yields Supremum of Parameters

Let $\struct {S, \preceq}$ be a totally ordered set.  Let $x, y \in S$. Then: :$\max \set {x, y} = \sup \set {x, y}$ where: :$\max$ denotes the max operation :$\sup$ denotes the supremum.


Mappings to Vector Space form Vector Space

Let $X$ be a non-empty set. Let $V$ be a vector space over a field (or division ring) $K$. Let $V^X$ denote the set of all mappings from $X$ to $V$. Let $+$ denote pointwise addition on $V^X$. Let $\circ$ denote pointwise ($K$)-scalar multiplication on $V^X$. Then $\left({V^X, +, \circ}\right)_K$ is a vector space over $K$.


Mappings to Algebraic Structure form Similar Algebraic Structure

Let $X$ be a nonempty set. Let $G$ be a magma with respect to the binary operations $\circ_1, \ldots, \circ_n$ on $G$. Let $G^X$ be the set of all mappings from $X$ to $G$. Denote also by $\circ_1, \ldots, \circ_n$ the binary operations defined on $G^X$ by pointwise addition. {{wtd|Transcluded pages for group, monoid, abelian group, and so on, and so forth}}


Mappings to R-Algebraic Structure form Similar R-Algebraic Structure

Let $X$ be a nonempty set, and let $R$ be a ring. Let $\left({G, \circ}\right)_R$ be an $R$-algebraic structure. Let $G^X$ be the set of all mappings from $X$ to $G$. Denote also by $\circ$ the binary operation defined on $G^X$ by pointwise ($R$)-scalar multiplication. {{wtd|Transcluded pages for module, vector space, and so on, and so forth}}


Space of Bounded Linear Transformations is Banach Space

Let $H, K$ be Hilbert spaces. Let $\map B {H, K}$ denote the space of bounded linear transformations from $H$ to $K$. Let $\Bbb F \in \set {\R, \C}$ denote the ground field of $K$. Now $\map B {H, K} \subseteq K^H$, the set of mappings from $H$ to $K$. Therefore, $\map B {H, K}$ can be endowed with pointwise addition ($+$) and ($\Bbb F$)-scalar multiplication ($\circ$). Let $\norm{\,\cdot\,}$ denote the norm on bounded linear transformations. Then $\norm{\,\cdot\,}$ is a norm on $\map B {H, K}$. Furthermore, $B \left({H, K}\right)$ is a Banach space with respect to this norm.


Equivalence of Formulations of Pasch's Axiom

The two forms of Pasch's Axiom in Tarski's Geometry are consistent. That is, the expressions: :$(1): \quad \forall a, b, c, p, q: \exists x: \mathsf B a p c \land \mathsf B b q c \implies \mathsf B p x b \land \mathsf B q x a$ and: :$(2): \quad \forall a, b, c, p, q: \exists x: \mathsf B a p c \land \mathsf B q c b \implies \mathsf B a x q \land \mathsf B b p x$ are logically equivalent.


Square-Summable Indexed Sets Closed Under Addition

Let $\family {a_i}_{i \mathop \in I}, \family {b_i}_{i \mathop \in I}$ be $I$-indexed families of real numbers. Let: :$\displaystyle \sum \set {a_i^2: i \in I} < \infty$ :$\displaystyle \sum \set {b_i^2: i \in I} < \infty$ where $\displaystyle \sum$ denotes the generalized sums. Then: :$\displaystyle \sum \set {\paren {a_i + b_i}^2: i \in I} < \infty$


Generalized Sum Preserves Inequality

Let $\left({a_i}\right)_{i \in I}, \left({b_i}\right)_{i \in I}$ be $I$-indexed families of positive real numbers. That is, let $a_i, b_i \in \R_{\ge 0}$ for all $i \in I$. Suppose that for all $i \in I$, $a_i \le b_i$. Furthermore, suppose that $\displaystyle \sum \left\{{ b_i: i \in I }\right\}$ converges. Then: : $\displaystyle \sum \left\{{ a_i: i \in I }\right\} \le \sum \left\{{ b_i: i \in I }\right\}$ In particular, $\displaystyle \sum \left\{{ a_i: i \in I }\right\}$ converges.


Generalized Sum is Linear

Let $\left({z_i}\right)_{i \in I}, \left({w_i}\right)_{i \in I}$ be $I$-indexed families of complex numbers. That is, let $z_i, w_i \in \C$ for all $i \in I$. Suppose that $\displaystyle \sum \left\{{ z_i: i \in I }\right\}, \sum \left\{{ w_i: i \in I }\right\}$ converge to $z, w \in \C$, respectively. Then: :$(1): \quad \displaystyle \sum \left\{{ z_i + w_i: i \in I }\right\}$ converges to $z+w$ :$(2): \quad \forall \lambda \in \C: \displaystyle \sum \left\{{ \lambda z_i: i \in I }\right\}$ converges to $\lambda z$


Min yields Infimum of Parameters

Let $\left({S, \preceq}\right)$ be a totally ordered set.  Let $x, y \in S$. Then: : $\min \left({x, y}\right) = \inf \left({\left\{{x, y}\right\}}\right)$ where: :$\min$ denotes the min operation :$\inf$ denotes the infimum.


Convergence of Generalized Sum of Complex Numbers

Let $\left({z_i}\right)_{i \mathop \in I}$ be an $I$-indexed family of complex numbers. That is, let $z_i \in \C$ for all $i \in I$. Denote by $\operatorname{Re} z_i$, resp. $\operatorname{Im} z_i$ the families of real, resp. imaginary parts of the family $z_i$. Then the following are equivalent: :$(1): \quad \displaystyle \sum \left\{{z_i : i \in I}\right\}$ converges to $z \in \C$ :$(2): \quad \displaystyle \sum \left\{{\operatorname{Re} z_i : i \in I}\right\}, \sum \left\{{\operatorname{Im} z_i : i \in I}\right\}$ converge to $\operatorname{Re} z, \operatorname{Im} z \in \R$, respectively


Lower and Upper Bounds for Sequences/Corollary

Let $\sequence {x_n}, \sequence {y_n}$ be sequences in $\R$. Let $x_n \to l, y_n \to m$ as $n \to \infty$. Suppose that for all $n \in \N$, $x_n \le y_n$. Then: :$l \le m$ that is: :$\displaystyle \lim_{n \mathop \to \infty} x_n \le \lim_{n \mathop \to \infty} y_n$ This is often phrased as: ''limits preserve inequalities''.


Bounded Linear Transformation Induces Bounded Sesquilinear Form

Let $H, K$ be Hilbert spaces over $\Bbb F \in \left\{{\R, \C}\right\}$. Let $A \in B \left({H, K}\right), B \in B \left({K, H}\right)$ be bounded linear transformations. Let $u, v: H \times K \to \Bbb F$ be defined by: :$u \left({h, k}\right) := \left\langle{Ah, k}\right\rangle_K$ :$v \left({h, k}\right) := \left\langle{h, Bk}\right\rangle_H$ Then $u$ and $v$ are bounded sesquilinear forms.


Classification of Bounded Sesquilinear Forms

Let $H, K$ be Hilbert spaces over $\Bbb F \in \left\{{\R, \C}\right\}$. Let $u: H \times K \to \Bbb F$ be a bounded sesquilinear form with bound $M$. Then there exist unique bounded linear transformations $A \in B \left({H, K}\right), B \in B \left({K, H}\right)$ such that: :$\forall h \in H, k \in K: u \left({h, k}\right) = \left\langle{Ah, k}\right\rangle_K = \left\langle{h, Bk}\right\rangle_H$ Furthermore, $\left\Vert{A}\right\Vert, \left\Vert{B}\right\Vert \le M$.


Isomorphism iff Inverse Equals Adjoint

Let $H, K$ be Hilbert spaces. Let $U \in B \struct {H, K}$ be a bounded linear transformation. Then the following are equivalent: :$(1): \quad U$ is an isomorphism :$(2): \quad U$ is invertible and $U^{-1} = U^*$, where $U^*$ denotes the adjoint of $U$.


Limit of Sequence is Limit of Real Function

Let $\sequence {a_n}$ be a real sequence. Let $f: x \mapsto \map f x$ be a real function. Suppose the limit: :$\displaystyle \lim_{x \mathop \to +\infty} \map f x$ exists. If for every $n$ in the domain of $\sequence {a_n}$: :$\map f n = a_n$ then: :$\displaystyle \lim_{n \mathop \to +\infty} \ a_n = \displaystyle \lim_{x \mathop \to +\infty} \map f x$


Equality of Cartesian Products

Let $A, B, C, D$ be nonempty sets. Then: : $A \times B = C \times D \iff A = C \land B = D$ where $\times$ denotes cartesian product.


Adjoining is Linear

Let $H, K$ be Hilbert spaces over $\Bbb F \in \left\{{\R, \C}\right\}$. Let $A, B \in B \left({H, K}\right)$ be bounded linear transformations. Then the operation of adjoining $^*$ satisfies, for all $\lambda \in \Bbb F$: :$(1): \qquad \left({\lambda A}\right)^* = \overline \lambda A^*$ :$(2): \qquad \left({A + B}\right)^* = A^* + B^*$ That is, $^*: B \left({H, K}\right) \to B \left({K, H}\right)$ is a linear transformation.


Adjoint of Composition

Let $H, K, L$ be Hilbert spaces. Let $A \in \map B {K, L}, B \in \map B {H, K}$ be bounded linear transformations. Then $\paren {A B}^* = B^* A^*$, where $^*$ denotes adjoining.


Double Adjoint is Itself

Let $H, K$ be Hilbert spaces. Let $A \in B \left({H, K}\right)$ be a bounded linear transformation. Then $A^{**} := \left({A^*}\right)^* = A$.


Identity of Points

Two points share the same position {{iff}} they are the same points.


Successor of Omega

:$\omega + 1 = \set {0, 1, 2, ...; \omega}$ where $\omega$ is the minimal infinite successor set and $\omega + 1$ is the successor of $\omega$. Note the use of the semicolon; this is the notation for multipart infinite sets.


Equivalence of Versions of Axiom of Choice

The following formulations of the Axiom of Choice are equivalent:


Norm of Adjoint

Let $H, K$ be Hilbert spaces. Let $A \in \map B {H, K}$ be a bounded linear transformation. Then the norm of $A$ satisfies: :$\norm A^2 = \norm {A^*}^2 = \norm {A^* A}$ where $A^*$ denotes the adjoint of $A$.


Operator Self-Adjoint iff Inner Product Real

Let $H$ be a Hilbert space over $\C$. Let $A \in \map B H$ be a bounded linear operator. Then $A$ is self-adjoint {{iff}}: :$\forall h \in H: \innerprod {A h} h_H \mathop \in \R$


Norm of Self-Adjoint Operator

Let $H$ be a Hilbert space. Let $A \in B \left({H}\right)$ be a self-adjoint operator. Then the norm of $A$ satisfies: :$\left\Vert{A}\right\Vert = \sup \left\{{ \left\vert{ \left\langle{Ah, h}\right\rangle_H }\right\vert: h \in H, \left\Vert{h}\right\Vert_H = 1 }\right\}$


Operator Zero iff Inner Product Zero

Let $H$ be a Hilbert space over $\C$. Let $A \in B \left({H}\right)$ be a bounded linear operator. Suppose that: :$\forall h \in H: \left\langle{Ah, h}\right\rangle_H = 0$ Then $A$ is the zero operator.


Linear Operator is Sum of Real and Imaginary Parts

Let $H$ be a Hilbert space over $\C$. Let $A \in B \left({H}\right)$ be a bounded linear operator. Let $B$ and $C$ be the real and imaginary parts of $A$, respectively. Then $A = B + iC$.


Characterization of Normal Operators

Let $H$ be a Hilbert space. Let $A \in B \left({H}\right)$ be a bounded linear operator. Then the following are equivalent: :$(1): \qquad AA^* = A^*A$, i.e., $A$ is normal :$(2): \qquad \forall h \in H: \left\Vert{Ah}\right\Vert_H = \left\Vert{A^*h}\right\Vert_H$ If $H$ is a Hilbert space over $\C$, these are also equivalent to: :$(3): \qquad \operatorname{Re} \left({A}\right) \operatorname{Im} \left({A}\right) = \operatorname{Im} \left({A}\right) \operatorname{Re} \left({A}\right)$, i.e., the real and imaginary parts of $A$ commute


Isometry iff Adjoint is Left-Inverse

Let $H, K$ be Hilbert spaces. Let $A \in B \left({H, K}\right)$ be a bounded linear transformation. Then $A$ is an isometry iff: :$A^*A = I_H$ where $A^*$ denotes the adjoint of $A$, and $I_H$ the identity operator on $H$.


Characterization of Unitary Operators

Let $H$ be a Hilbert space. Let $A \in \map B H$ be a bounded linear operator. Then the following are equivalent: :$(1): \quad A$ is a unitary operator :$(2): \quad A^* A = A A^* = I$, where $A^*$ denotes the adjoint of $A$, and $I$ denotes the identity operator :$(3): \quad A$ is a normal isometry


Kernel of Linear Transformation is Orthocomplement of Range of Adjoint

Let $H, K$ be Hilbert spaces. Let $A \in B \struct {H, K}$ be a bounded linear transformation. Then $\ker A = \paren {\Img {A^*} }^\perp$, where: :$A^*$ denotes the adjoint of $A$ :$\ker A$ is the kernel of $A$ :$\Img {A^*}$ is the image of $A^*$ :$\perp$ signifies orthocomplementation


Relation Contains Mapping is Equivalent to AoC

Let $S$ and $T$ be sets. Let $\RR \subseteq S \times T$ be a relation on $S \times T$. Then: :there exists a mapping $f \subseteq \RR$ whose domain is the same as the preimage of $\RR$ {{iff}} :the axiom of choice holds.


Existence of Set with Singleton Intersections with Disjoint Collection

Let $\mathcal C$ be a set of sets all of which are pairwise disjoint. Then: : there exists a set $A$ such that $\forall S \in \mathcal C: A \cap S$ is a singleton {{iff}} : the axiom of choice holds.


Complementary Idempotent is Idempotent

Let $H$ be a Hilbert space. Let $A$ be an idempotent operator. Then the complementary idempotent $I - A$ is also idempotent.


Range of Idempotent is Kernel of Complementary Idempotent

Let $H$ be a Hilbert space. Let $A$ be an idempotent operator. Then $\Rng A = \map \ker {I - A}$.


Range and Kernel of Idempotent are Algebraically Complementary

Let $H$ be a Hilbert space. Let $A$ be an idempotent operator. Then $\ker A$ and $\Rng A$ are algebraically complementary, that is: :$\ker A \cap \Rng A = \left({0}\right)$, the zero subspace :$\ker A + \Rng A = H$, where $+$ signifies setwise addition. {{explain|Determine whether range means image or codomain, and replace appropriately}}


Natural Logarithm Function is Continuous

The natural logarithm function is continuous.


Characterization of Projections

Let $H$ be a Hilbert space. Let $A \in B \left({H}\right)$ be an idempotent operator. Then the following are equivalent: :$(1): \qquad A$ is a projection :$(2): \qquad A$ is the orthogonal projection onto $\operatorname{ran} A$ :$(3): \qquad \left\Vert{A}\right\Vert = 1$, where $\left\Vert{\cdot}\right\Vert$ is the norm on bounded linear operators. :$(4): \qquad A$ is self-adjoint :$(5): \qquad A$ is normal :$(6): \qquad \forall h \in H: \left\langle{Ah, h}\right\rangle_H \ge 0$


Set of Linear Subspaces is Closed under Intersection

Let $\struct {V, +, \circ}_K$ be a $K$-vector space. Let $\family {M_i}_{i \mathop \in I}$ be an $I$-indexed family of subspaces of $V$. Then $M := \ds \bigcap_{i \mathop \in I} M_i$ is also a subspace of $V$.


Closed Linear Subspaces Closed under Intersection

Let $V$ be a topological vector space. Let $\family {M_i}_{i \mathop \in I}$ be an $I$-indexed family of closed linear subspaces of $V$. Then $M := \ds \bigcap_{i \mathop \in I} M_i$ is also a closed linear subspace of $V$.


Orthogonal Difference is Closed Linear Subspace

Let $H$ be a Hilbert space. Let $M, N$ be closed linear subspaces of $H$. Then the orthogonal difference $M \ominus N$ is also a closed linear subspace of $H$.


Closed Linear Subspaces Closed under Setwise Addition

Let $H$ be a Hilbert space. Let $M, N$ be closed linear subspaces of $H$. Then $M + N$ is also a closed linear subspace of $H$, where $+$ denotes setwise addition.


Linear Subspaces Closed under Setwise Addition

Let $V$ be a $K$-vector space. Let $M, N$ be linear subspaces of $V$. Then $L := M + N$ is also a linear subspace of $V$, where $+$ denotes setwise addition.


Complementary Projection is Projection

Let $H$ be a Hilbert space. Let $A$ be a projection. Then the complementary projection $I - A$ is also a projection.


Direct Sum of Subspace and Orthocomplement

Let $H$ be a Hilbert space. Let $M$ be a closed linear subspace of $H$. Denote by $M^\perp$ its orthocomplement. Then the direct sum $M \oplus M^\perp$ is isomorphic to $H$.


Characterization of Invariant Subspaces

Let $H$ be a Hilbert space. Let $A \in B \left({H}\right)$ be a bounded linear operator. Let $\begin{pmatrix}  W & X \\  Y & Z \end{pmatrix}$ be the matrix notation for $A$ with respect to $M$. Let $M$ be a closed linear subspace of $H$; denote by $P$ the orthogonal projection on $M$. Then the following three statements are equivalent: :$(1): \qquad M$ is an invariant subspace for $A$ :$(2): \qquad PAP = AP$ :$(3): \qquad Y = 0$


Characterization of Reducing Subspaces

Let $\HH$ be a Hilbert space. Let $A \in \map B \HH$ be a bounded linear operator. Let $M$ be a closed linear subspace of $\HH$. Let $P$ denote the orthogonal projection on $M$. Let $\begin{pmatrix}  W & X \\  Y & Z \end{pmatrix}$ be the matrix notation for $A$ with respect to $M$. Then the following four statements are equivalent: :$(1): \quad M$ is a reducing subspace for $A$ :$(2): \quad P A = A P$ :$(3): \quad X = Y = 0$ :$(4): \quad M$ is an invariant subspace for both $A$ and its adjoint $A^*$


Subset of Toset is Toset

Let $\left({S, \preceq}\right)$ be a totally ordered set. Let $T \subseteq S$. Then $\left({T, \preceq \restriction_T}\right)$ is also a totally ordered set. In the above, $\preceq \restriction_T$ denotes the restriction of $\preceq$ to $T$.


Bretschneider's Formula

Let $ABCD$ be a general quadrilateral. Then the area $\mathcal A$ of $ABCD$ is given by: :$\mathcal A = \sqrt{\left({s - a}\right) \left({s - b}\right) \left({s - c}\right) \left({s - d}\right) - a b c d \cos^2 \left({\dfrac {\alpha + \gamma} 2}\right)}$ where: : $a, b, c, d$ are the lengths of the sides of the quadrilateral : $s = \dfrac {a + b + c + d} 2$ is the semiperimeter : $\alpha$ and $\gamma$ are opposite angles.


Transfinite Induction/Principle 1

Let $\On$ denote the class of all ordinals. Let $A$ denote a class. Suppose that: :For all elements $x$ of $\On$, if $x$ is a subset of $A$, then $x$ is an element of $A$. Then $\On \subseteq A$.


Limit of Functions that Agree

Let $f$ and $g$ be real functions. Let $f$ and $g$ agree for all $x$ in a deleted neighborhood of $c$. Let the limit: :$\displaystyle \lim_{x \to c} \ f \left({x}\right)$ exist. Then the limit: :$\displaystyle \lim_{x \to c} \ g \left({x}\right)$ also exists, and: :$\displaystyle \lim_{x \to c} \ f \left({x}\right) =  \lim_{x \to c} \ g \left({x}\right)$


Ordering of Reciprocals

Let $x, y \in \R$ be real numbers such that $x, y \in \openint 0 \to$ or $x, y \in \openint \gets 0$ Then: :$x \le y \iff \dfrac 1 y \le \dfrac 1 x$


Comparison Test for Divergence

Let $\displaystyle \sum_{n \mathop = 1}^\infty b_n$ be a divergent series of positive real numbers. Let $\sequence {a_n}$ be a sequence in $\R$. Let: :$\forall n \in \N_{>0}: b_n \le a_n$ Then the series $\displaystyle \sum_{n \mathop = 1}^\infty a_n$ diverges.


Limit at Infinity of Real Identity Function

Let $I_\R: \R \to \R$ be the identity function on $\R$. Then: :$(1): \quad \displaystyle \lim_{x \mathop \to +\infty} \map {I_\R} x = +\infty$ :$(2): \quad \displaystyle \lim_{x \mathop \to -\infty} \map {I_\R} x = -\infty$


Push Theorem

Let $f$ be a real function which is continuous on the open interval $\openint a \to$, $a \in \R$, such that: :$\displaystyle \lim_{x \mathop \to +\infty} \map f x = +\infty$ Let $g$ be a real function defined on some open interval $\openint b \to$ such that, for sufficiently large $x$: :$\map g x > \map f x$ Then: :$\displaystyle \lim_{x \mathop \to +\infty} \map g x = +\infty$


Orthocomplement Reverses Subset

Let $H$ be a Hilbert space. Let $A, B$ be subsets of $H$, and let $A \subseteq B$. Then $B^\perp \subseteq A^\perp$, where $\perp$ signifies orthocomplementation.


Limit at Infinity of x^n

Let $x \mapsto x^n$, $n \in \R$ be a real function which is continuous on the open interval $\openint 1 {+\infty}$. Let $n > 0$. Then $x^n \to +\infty$ as $x \to +\infty$.


Monotonicity of Real Sequences

Let $\sequence {a_n}: \mathbb D \to \R$ be a real sequence, where $\mathbb D$ is a subset of $\N$. Let $\Bbb X$ be a real interval such that $\Bbb D \subseteq \Bbb X$. Let $f: \Bbb X \to \R, x \mapsto \map f x$ be a differentiable real function. Suppose that for every $n \in \mathbb D$: :$\map f n = a_n$ Then: :If $\forall x \in \Bbb X: D_x \map f x \ge 0$, $\sequence {a_n}$ is increasing :If $\forall x \in \Bbb X: D_x \map f x >  0$, $\sequence {a_n}$ is strictly increasing :If $\forall x \in \Bbb X: D_x \map f x \le 0$, $\sequence {a_n}$ is decreasing :If $\forall x \in \Bbb X: D_x \map f x < 0$, $\sequence {a_n}$ is strictly decreasing where $D_x$ denotes differentiation {{WRT|Differentiation}} $x$.


Hausdorff Maximal Principle is equivalent to Axiom of Choice

Every ordered set has a maximal chain {{iff}} the axiom of choice holds.


Young's Inequality for Products

Let $p, q \in \R_{> 0}$ be strictly positive real numbers such that: : $\dfrac 1 p + \dfrac 1 q = 1$ Then, for any $a, b \in \R_{\ge 0}$: : $a b \le \dfrac {a^p} p + \dfrac{b^q} q$ Equality occurs {{iff}}: :$b = a^{p-1}$.


Restriction of Monotone Function is Monotone

The restriction of a monotone mapping is monotone.


Taylor's Theorem/One Variable/Statement of Theorem

Let $f$ be a real function which is: :of differentiability class $C^n$ on the closed interval $\closedint a x$ and: :at least $n + 1$ times differentiable on the open interval $\openint a x$. Then: {{begin-eqn}} {{eqn | l = \map f x       | r = \frac 1 {0!} \map f a       | c = }} {{eqn | o =       | ro= +       | r = \frac 1 {1!} \paren {x - a} \map {f'} a       | c =  }} {{eqn | o =       | ro= +       | r = \frac 1 {2!} \paren {x - a}^2 \map {f''} a       | c =  }} {{eqn | o =       | ro= +       | r = \cdots       | c =  }} {{eqn | o =       | ro= +       | r = \frac 1 {n!} \paren {x - a}^n \map {f^{\paren n} } a       | c =  }} {{eqn | o =       | ro= +       | r = R_n       | c =  }} {{end-eqn}} where $R_n$ (sometimes denoted $E_n$) is known as the '''error term''' or '''remainder''', and can be presented in one of $2$ forms: ;Lagrange Form :$R_n = \dfrac {\map {f^{\paren {n + 1} } } \xi} {\paren {n + 1}!} \paren {x - a}^{n + 1}$ for some $\xi \in \openint a x$. ;Cauchy Form :$R_n = \dfrac {\map {f^{\paren {n + 1} } } \xi \paren {x - \xi}^n} {n!} \paren {x - a}$ for some $\xi \in \openint a x$.


Elements of Minimal Infinite Successor Set are Well-Ordered

Let $\omega$ be the minimal infinite successor set. Let $a \in \omega$. Then $a$ is well-ordered by $\subseteq$.


Finite Lexicographic Order on Well-Ordered Sets is Well-Ordering

Let $S$ be a set which is well-ordered by $\preceq$. Let $\preccurlyeq$ be the lexicographic order on the set $T_n$ of all ordered $n$-tuples of $S$: :$\left({x_1, x_2, \ldots, x_n}\right) \prec \left({y_1, y_2, \ldots, y_n}\right)$ {{iff}}: ::$\exists k: 1 \le k \le n$ such that $\forall 1 \le j < k: x_j = y_j$ but $x_k \prec y_k$ in $S$. {{explain|Disentangle this.}} Then for a given $n \in \N_{>0}$, $\preccurlyeq$ is a well-ordering on $T_n$.


Infinite Lexicographic Order on Well-Ordered Sets is not Well-Ordering

Let $\struct {S, \preceq}$ be a well-ordered set. Let $\preccurlyeq$ be the lexicographic order on the set $T$ of all ordered tuples of $S$: :$\tuple {x_1, x_2, \ldots, x_m} \prec \tuple {y_1, y_2, \ldots, y_n}$ {{iff}}: ::$\exists k: 1 \le k \le \map \min {m, n}$ such that $\forall 1 \le j < k: x_j = y_j$ but $x_k \prec y_k$ in $S$ :or: ::$m < n$ and $\forall 1 \le j < m: x_j = y_j$. Then $\preccurlyeq$ is '''not''' a well-ordering on $T$.


Minimal Infinite Successor Set is Well-Ordered

Let $\omega$ be the minimal infinite successor set. Then $\omega$ is well-ordered by $\subseteq$.


Transfinite Induction

=== Principle 1 === {{:Transfinite Induction/Principle 1}} === Schema 1 === {{:Transfinite Induction/Schema 1}} === Principle 2 === {{:Transfinite Induction/Principle 2}} === Schema 2 === {{:Transfinite Induction/Schema 2}}


Transfinite Recursion/Theorem 1

Let $G$ be a mapping. {{explain|What are the domain and range of $G$?}} Let $K$ be a class of mappings $f$ that satisfy: :the domain of $f$ is some ordinal $y$ :$\forall x \in y: f \left({x}\right) = G \left({f {\restriction_x} }\right)$ where $f {\restriction_x}$ denotes the restriction of $f$ to $x$. Let $F = \bigcup K$, the union of $K$. Then: :$(1): \quad F$ is a mapping with domain $\operatorname{On}$ :$(2): \quad \forall x \in \operatorname{On}: F \left({x}\right) = G \left({F {\restriction_x} }\right)$ :$(3): \quad F$ is unique. That is, if another mapping $A$ has the above two properties, then $A = F$.


Squeeze Theorem for Absolutely Convergent Series

Let $\displaystyle \sum \size {a_n}$ be an absolutely convergent series in $\R$. Suppose that: :$\displaystyle -\sum \size {a_n} = \sum \size {a_n}$ Then $\displaystyle \sum a_n$ equals the above two series.


N Choose k is not greater than n^k

:$\forall n \in \Z, k \in \Z: 1 \le k \le n: \dbinom n k < n^k$  where $\dbinom n k$ is a binomial coefficient. Equality holds when $k = 0$ and $k = 1$.


Absolutely Convergent Series is Convergent/Real Numbers

Let $\displaystyle \sum_{n \mathop = 1}^\infty a_n$ be an absolutely convergent series in $\R$. Then $\displaystyle \sum_{n \mathop = 1}^\infty a_n$ is convergent.


Definition:Natural Numbers/Zermelo Construction

The natural numbers $\N = \set {0, 1, 2, 3, \ldots}$ can be defined as a series of subsets: :$0 := \O = \set {}$ :$1 := \set 0 = \set \O$ :$2 := \set 1 = \set {\set \O}$ :$3 := \set 2 = \set {\set {\set \O} }$ ::::$\vdots$ Thus the natural number $n$ consists of $\O$ enclosed in $n$ pairs of braces.


Set is Element of its Power Set

A set is an element of its power set: :$S \in \powerset S$


Power Set of Empty Set

The power set of the empty set $\O$ is the set $\set \O$.


Symmetric Difference is Subset of Union

The symmetric difference of two sets is a subset of their union: :$S * T \subseteq S \cup T$


Sum of Projections/General Case

Let $H$ be a Hilbert space. Let $\left({M_i}\right)_{i \in I}$ be an $I$-indexed set of closed linear subspaces of $H$. Let $M_i$ and $M_j$ be orthogonal whenever $i \ne j$. Denote, for each $i \in I$, by $P_i$ the orthogonal projection onto $M_i$. Denote by $P$ the orthogonal projection onto the closed linear span $\vee \left\{{M_i: i \in I}\right\}$ of the $M_i$. Then for all $h \in H$, $\displaystyle \sum \left\{{P_i h: i \in I}\right\} = Ph$, where $\displaystyle \sum$ denotes a generalized sum.


Product of Projections

Let $H$ be a Hilbert space. Let $P, Q$ be projections. Then the following are equivalent: :$(1): \quad PQ$ is a projection :$(2): \quad PQ = QP$ :$(3): \quad P + Q - PQ$ is a projection {{MissingLinks|Provide proper linking to the def of addition and multiplication of operators}}


Difference of Projections

Let $H$ be a Hilbert space. Let $P, Q$ be projections. Then the following are equivalent: :$(1): \quad P - Q$ is a projection :$(2): \quad P Q = Q$ :$(3): \quad Q P = Q$ {{MissingLinks|Provide proper linking to the def of addition of operators}}


Union of Countable Sets of Sets

Let $\mathcal A$ and $\mathcal B$ be countable sets of sets. Then: :$\left\{{A \cup B: A \in \mathcal A, B \in \mathcal B}\right\}$ is also countable.


Set of Finite Subsets of Countable Set is Countable

Let $A$ be a countable set. Then the set of finite subsets of $A$ is countable.


Cantor's Theorem (Strong Version)

Let $S$ be a set. Let $\map {\PP^n} S$ be defined recursively by: :$\map {\PP^n} S = \begin{cases} S & : n = 0 \\ \powerset {\map {\PP^{n - 1} } S} & : n > 0 \end{cases}$ where $\powerset S$ denotes the power set of $S$. Then $S$ is not equivalent to $\map {\PP^n} S$ for any $n > 0$.


Ordering Induced by Injection is Ordering

Let $\left({T, \le}\right)$ be an ordered set, and let $S$ be a set. Let $f: S \to T$ be an injection. Then $\le_f$, the ordering induced by $f$, is an ordering.


Injection Induces Total Ordering

Let $\left({T, \le}\right)$ be a totally ordered set, and let $S$ be a set. Let $f: S \to T$ be an injection. Then $\le_f$, the ordering induced by $f$, is a total ordering.


Injection Induces Well-Ordering

Let $\left({T, \le}\right)$ be a well-ordered set. Let $S$ be a set. Let $f: S \to T$ be an injection. Then $\le_f$, the ordering induced by $f$, is a well-ordering.


Hartogs' Lemma (Set Theory)

Let $S$ be a set. Then there exists an ordinal $\alpha$ such that there is no injection from $\alpha$ to $S$.


Bourbaki-Witt Fixed Point Theorem

Let $\struct {X, \le}$ be a non-empty chain complete poset (that is, an ordered set in which every chain has a supremum). Let $f: X \to X$ be an inflationary mapping, that is, so that $\map f x \ge x$. Then for every $x \in X$ there exists $y \in X$ where $y \ge x$ such that $\map f y = y$.


Convergent Sequence with Finite Number of Terms Deleted is Convergent

Let $\left({X, d}\right)$ be a metric space. Let $\left\langle{x_k}\right\rangle$ be a sequence in $X$. Let $\left\langle{x_k}\right\rangle$ be convergent. Let a finite number of terms be deleted from $\left \langle {x_k} \right \rangle$. Then the resulting subsequence is convergent.


Cauchy Condensation Test

Let $\sequence {a_n}: n \mapsto \map a n$ be a decreasing sequence of strictly positive terms in $\R$ which converges with a limit of zero. That is, for every $n$ in the domain of $\sequence {a_n}$: $a_n > 0$, $a_n \ge a_{n + 1}$, and $a_n \to 0$ as $n \to +\infty$. Then the series: :$\displaystyle \sum_{n \mathop = 1}^\infty a_n$  converges {{iff}} the condensed series: :$\displaystyle \sum_{n \mathop = 1}^\infty 2^n \map a {2^n}$ converges.


Between Two Sets Exists Injection or Surjection

Let $S$ and $T$ be sets. Then either or both of the following cases hold: :$(1):$ There exists a mapping $f: S \to T$ such that $f$ is an injection :$(2):$ There exists a mapping $f: S \to T$ such that $f$ is a surjection.


Injection from Set to Power Set

For every set $S$, there exists an injection from $S$ to its power set $\mathcal P \left({S}\right)$.


Minimum Degree is at Least Connectivity

Let $G = \left({V, E}\right)$ be a simple graph. Then: :$\delta \left({G}\right) \geq \kappa \left({G}\right)$ That is, the minimum degree of $G$ is at least its connectivity.


Homeomorphism Relation is Equivalence

Let $T_1$ and $T_2$ be topological spaces. Let $T_1 \sim T_2$ denote that $T_1$ and $T_2$ are homeomorphic. The relation $\sim$ is an equivalence relation.


Identity Mapping is Homeomorphism

Let $T$ be a topological space. The identity mapping $I_T: T \to T$ defined as: :$\forall x \in T: I_T \left({x}\right) = x$ is a homeomorphism.


Composite of Homeomorphisms is Homeomorphism

Let $T_1, T_2, T_3$ be topological spaces.  Let $f: T_1 \to T_2$ and $g: T_2 \to T_3$ be homeomorphisms. Then $g \circ f: T_1 \to T_3$ is also a homeomorphism.


Identity Mapping is Continuous

Let $T = \struct {S, \tau}$ be a topological space. The identity mapping $I_S: S \to S$ defined as: :$\forall x \in S: \map {I_S} x = x$ is a continuous mapping.


Kuratowski's Lemma

Let $\struct {S, \preceq}, S \ne \O$ be a non-empty ordered set.  Then every chain in $S$ is the subset of some maximal chain.


Included Set Topology is Topology

Let $T = \left({S, \tau_H}\right)$ be an included set space. Then $\tau_H$ is a topology on $S$, and $T$ is a topological space.


Space of Compact Linear Transformations is Banach Space

Let $H, K$ be Hilbert spaces, and let $B_0 \left({H, K}\right)$ be the space of compact linear transformations from $H$ to $K$. Let $\Bbb F \in \left\{{\R, \C}\right\}$ be the ground field of $K$. Now $B_0 \left({H, K}\right) \subseteq K^H$, the set of mappings from $H$ to $K$. Therefore, $B_0 \left({H, K}\right)$ can be endowed with pointwise addition ($+$) and ($\Bbb F$)-scalar multiplication ($\circ$). Let $\left\Vert{\cdot}\right\Vert$ denote the norm on bounded linear operators. Then $\left\Vert{\cdot}\right\Vert$ is a norm on $B_0 \left({H, K}\right)$. Furthermore, $B_0 \left({H, K}\right)$ is a Banach space with respect to this norm.


Compact Linear Transformation is Bounded

Let $H, K$ be Hilbert spaces. Let $T \in B_0 \left({H, K}\right)$ be a compact linear transformation. Then $T$ is also a bounded linear transformation. That is, $B_0 \left({H, K}\right) \subseteq B \left({H, K}\right)$.


Compact Linear Transformations Composed with Bounded Linear Operator

Let $H, K$ be Hilbert spaces. Let $T \in B_0 \left({H, K}\right)$ be a compact linear transformation. Let $A \in B \left({H}\right), B \in B \left({K}\right)$ be bounded linear operators. Then the compositions $TA$ and $BT$ are also compact linear transformations.


Linear Transformation Compact iff Adjoint Compact

Let $H, K$ be Hilbert spaces. Let $T: H \to K$ be a linear transformation. Then $T$ is compact iff its adjoint $T^*$ is.


Finite Rank Operators Dense in Compact Linear Transformations

Let $H, K$ be Hilbert spaces. Then: :$\map {B_{00} } {H, K}$ is everywhere dense in $\map {B_0} {H, K}$ where: :$\map {B_{00} } {H, K}$ is the space of continuous finite rank operators from $H$ to $K$ :$\map {B_0} {H, K}$ is the space of compact linear transformations from $H$ to $K$. That is, for every $T \in \map {B_0} {H, K}$, there is a sequence $\sequence {T_n}_{n \mathop \in \N}$ in $\map {B_{00} } {H, K}$ such that: :$\ds \lim_{n \mathop \to \infty} \norm {T_n - T} = 0$ where $\norm {\, \cdot \,}$ denotes the norm on bounded linear transformations.


Inverse of Matrix Product

Let $\mathbf {A, B}$ be square matrices of order $n$ Let $\mathbf I$ be the $n \times n$ unit matrix. Let $\mathbf A$ and $\mathbf B$ be invertible. Then the matrix product $\mathbf {AB}$ is also invertible, and: :$\paren {\mathbf A \mathbf B}^{-1} = \mathbf B^{-1} \mathbf A^{-1}$


Closure of Range of Compact Linear Transformation is Separable

Let $H, K$ be Hilbert spaces. Let $T \in \map {B_0} {H, K}$ be a compact linear transformation. Then $\map \cl {\Rng T}$ is separable.


Elementary Row Operations as Matrix Multiplications

Let $e$ be an elementary row operation. Let $\mathbf E$ be the elementary row matrix of order $m$ defined as: :$\mathbf E = e \paren {\mathbf I}$ where $\mathbf I$ is the unit matrix. Then for every $m \times n$ matrix $\mathbf A$: :$e \paren {\mathbf A} = \mathbf {E A}$ where $\mathbf {E A}$ denotes the conventional matrix product.


Included Set Topology on Finite Intersection

Let $T = \struct {S, \tau}$ be a topological space on a set $S$. Let $A_1, A_2, \ldots, A_n$ be a finite set of subsets of $S$: :$\forall i \in \closedint 1 n: A_i \subseteq S$ Let $\forall i \in \closedint 1 n: \map T {A_i} = \struct {S, \tau_{A_i} }$ be the included set spaces on $S$ by $A_i$. Let: :$\forall i \in \closedint 1 n: \map T {A_i} \le T$ where $\map T {A_i} \le T$ denotes that $\map T {A_i}$ is coarser than $T$. Then: :$\displaystyle \map T {\bigcap A_i} \le T$ where $\displaystyle \map T {\bigcap A_i}$ is the included set space on $S$ by $\displaystyle \bigcap_{i \mathop = 1}^n A_i$.


Included Set Topology on Union

Let $T = \struct {S, \tau}$ be a topological space on a set $S$. Let $\family {A_i}_{i \mathop \in I}$ be a family of subsets of $S$ indexed by the indexing set $I$: :$\forall i \in I: A_i \subseteq S$ Let $\forall i \in I: \map T {A_i} = \struct {S, \tau_{A_i} }$ be the included set spaces on $S$ by $A_i$. Let: :$\forall i \in I: \map T {A_i} \ge T$ where $\map T {A_i} \ge T$ denotes that $\map T {A_i}$ is finer than $T$. Then: :$\map T {\bigcup A_i} \ge T$ where $\map T {\bigcup A_i}$ is the included set space on $S$ by $\displaystyle \bigcup_{i \mathop \in I} A_i$.


Kuratowski's Free Set Theorem

Let $n \in \Z_{>0}$ be a positive integer. Let $X$ be an infinite set. Let $\left[{X}\right]^n$ denote the set of $n$-element subsets of $X$ whose cardinality is $n$. Let $\left[{X}\right]^{< \aleph_0}$ denote the set of finitesubsets of $X$. Then: : the cardinality of $X$ is greater than or equal to $\aleph_n$ {{iff}} : for every mapping $f: \left[{X}\right]^n \to \left[{X}\right]^{<\aleph_0}$ there exists an $(n + 1)$-element free subset of $X$ with respect to $f$. That is, a subset $Y$ of $X$ with $n+1$ elements such that $y \notin f \left({Y \setminus \left\{{y}\right\} }\right)$ for all $y \in Y$.


Diagonalizable Operator Bounded iff Value Set Bounded

Let $H$ be a Hilbert space. Let $A: H \to H$ be a diagonalizable operator. Let $\left({\alpha_e}\right)_{e \in E}$ be the value set of $A$, with respect to a suitable basis $E$ for $H$. Then $A$ is bounded iff $\left({\alpha_e}\right)_{e \in E}$ is, i.e., iff: :$\exists M \in \R: \forall e \in E: \left\vert{\alpha_e}\right\vert \le M$


Diagonalizable Operator Compact iff Value Set Converges to Zero

Let $H$ be a Hilbert space of countable dimension. Let $A: H \to H$ be a diagonalizable operator. Let $\left({\alpha_n}\right)_{n \in \N}$ be the value set of $A$, with respect to a suitable basis $E = \left({e_n}\right)_{n \in \N}$ for $H$. Then $A$ is compact iff: :$\displaystyle \lim_{n \to \infty} \alpha_n = 0$


Nonzero Eigenvalue of Compact Operator has Finite Dimensional Eigenspace

Let $H$ be a Hilbert space. Let $T \in B_0 \left({H}\right)$ be a compact operator. Let $\lambda \in \sigma_p \left({T}\right), \lambda \ne 0$ be a nonzero eigenvalue of $T$. Then the eigenspace for $\lambda$ has finite dimension.


Condition for Nonzero Eigenvalue of Compact Operator

Let $H$ be a Hilbert space over $\Bbb F \in \set {\R, \C}$. Let $T \in \map {B_0} H$ be a compact operator. Let $\lambda \in \Bbb F, \lambda \ne 0$ be a nonzero scalar. Suppose that the following holds: :$\inf \set {\norm {\paren {T - \lambda I} h}_H: \norm h_H = 1} = 0$ Then $\lambda \in \map {\sigma_p} T$, that is, $\lambda$ is an eigenvalue for $T$.


Finite Rank Operator is Compact

Let $H, K$ be Hilbert spaces. Let $T \in \map {B_{00} } {H, K}$ be a bounded finite rank operator. Then $T \in \map {B_0} {H, K}$, that is, $T$ is compact.


Adjoint of Finite Rank Operator

Let $H, K$ be Hilbert spaces. Let $T \in \map {B_{00} } {H, K}$ be a bounded finite rank operator. Then: :$T^* \in \map {B_{00} } {K, H}$ that is, the adjoint of $T$ is also a bounded finite rank operator.


Compact Idempotent is of Finite Rank

Let $H$ be a Hilbert space. Let $T \in \map {B_0} H$ be a compact linear operator, and let $T$ be idempotent. Then: :$T \in \map {B_{00} } H$ that is, $T$ is a bounded finite rank operator.


Characterization of Finite Rank Operators

Let $H$ be a Hilbert space. Let $T \in B_{00} \left({H}\right)$ be a bounded finite rank operator. Let $n = \operatorname{dim} \left({\operatorname{ran} T}\right)$ be the rank of $T$. Then there are orthonormal vectors $e_1, \ldots, e_n$ and vectors $g_1, \ldots, g_n$ of $H$ such that: :$\forall h \in H: Th = \displaystyle \sum_{i=1}^n \left\langle{h, e_i}\right\rangle_H g_i$


Compact Operator on Hilbert Space Direct Sum

Let $\sequence {H_n}_{n \mathop \in \N}$ be a sequence of Hilbert spaces. Denote by $H = \displaystyle \bigoplus_{n \mathop = 1}^\infty H_n$ their Hilbert space direct sum. For each $n \in \N$, let $T_n \in \map B {H_n}$ be a bounded linear operator. Suppose that one has $\displaystyle \sup_{n \mathop \in \N} \norm {T_n} < \infty$, where $\norm {\, \cdot \, }$ signifies the norm on bounded linear operators. Define $T \in \map B H$ by:  :$\forall h = \sequence {h_n}_{n \mathop \in \N}: T h = \sequence {T_n h_n}_{n \mathop \in \N} \in H$ (That $T$ is indeed bounded follows from Bounded Linear Operator on Hilbert Space Direct Sum.) Then $T$ is compact {{iff}} the following conditions hold: : For each $n \in \N$, $T_n$ is compact : $\displaystyle \lim_{n \mathop \to \infty} \norm {T_n} = 0$


Bounded Linear Operator on Hilbert Space Direct Sum

Let $\left({H_n}\right)_{n \in \N}$ be a sequence of Hilbert spaces. Denote by $H = \displaystyle \bigoplus_{n \mathop = 1}^\infty H_n$ their Hilbert space direct sum. For each $n \in \N$, let $T_n \in B \left({H_n}\right)$ be a bounded linear operator. Suppose that one has $\displaystyle \sup_{n \mathop \in \N} \, \left\Vert{T_n}\right\Vert < \infty$, where $\left\Vert{\cdot}\right\Vert$ signifies the norm on bounded linear operators. Define $T: H \to H$ by:  :$\forall h = \left({h_n}\right)_{n \in \N}: T h = \left({T_n h_n}\right)_{n \in \N} \in H$ Then $T \in B \left({H}\right)$ is a bounded linear operator.


Compact Self-Adjoint Operator has Countable Point Spectrum

Let $H$ be a Hilbert space. Let $T \in B_0 \left({H}\right)$ be a compact, self-adjoint operator. Then its point spectrum $\sigma_p \left({T}\right)$ is countable.


Spectral Theorem for Compact Self-Adjoint Operators

Let $H$ be a Hilbert space. Let $T \in B_0 \left({H}\right)$ be a compact, self-adjoint operator. Then there exists a (possibly finite) sequence $\left({\lambda_n}\right)$ of distinct nonzero eigenvalues of $T$ such that: * $P_n P_m = P_m P_n = 0$ if $n \ne m$ * $\displaystyle \lim_{k \to \infty} \left\Vert{T - \sum_{n=1}^k \lambda_n P_n}\right\Vert = 0$, that is, $T = \displaystyle \sum_{n=1}^\infty \lambda_n P_n$ where $P_n$ is the orthogonal projection onto the eigenspace of $\lambda_n$, and $\left\Vert{\cdot}\right\Vert$ denotes the norm on bounded linear operators. {{refactor|Split corollaries to subpages}} === Corollary 1 === There exists a (possibly finite) sequence $\left({\mu_n}\right)$ of real numbers and a basis $E = \left({e_n}\right)$ for $\left({\operatorname{ker} T}\right)^\perp$ such that: :$\forall h \in H: Th = \displaystyle \sum_{n=1}^\infty \left\langle{h, e_n}\right\rangle_H \mu_n e_n$


Eigenspace for Normal Operator is Reducing Subspace

Let $H$ be a Hilbert space over $\Bbb F \in \left\{{\R, \C}\right\}$. Let $A \in B \left({H}\right)$ be a normal operator. Let $\lambda \in \Bbb F$. Then $\ker \left({A - \lambda}\right)$ is a reducing subspace for $A$. Here $\ker$ denotes kernel.


Eigenvalues of Self-Adjoint Operator are Real

Let $H$ be a Hilbert space. Let $A \in B \left({H}\right)$ be a self-adjoint operator. Then all eigenvalues of $A$ are real.


Operator Diagonalizable iff Basis of Eigenvectors

Let $H$ be a Hilbert space. Let $A:H \to H$ be a linear operator on $H$. Then $A$ is diagonalizable iff there exists a basis $E$ of $H$, consisting of eigenvectors for $A$.


Existence of Hartogs Number

Let $S$ be a set. Then $S$ has a Hartogs number.


Left and Right Inverse Mappings Implies Bijection

Let $f: S \to T$ be a mapping. Let $f$ be such that: : $\exists g_1: T \to S: g_1 \circ f = I_S$ : $\exists g_2: T \to S: f \circ g_2 = I_T$ where both $g_1$ and $g_2$ are mappings. Then $f$ is a bijection.


Left and Right Inverses of Mapping are Inverse Mapping

Let $f: S \to T$ be a mapping such that: :$(1): \quad \exists g_1: T \to S: g_1 \circ f = I_S$ :$(2): \quad \exists g_2: T \to S: f \circ g_2 = I_T$ Then: : $g_1 = g_2 = f^{-1}$ where $f^{-1}$ is the inverse of $f$.


Operator Commuting with Diagonalizable Operator

Let $H$ be a Hilbert space. Let $A = \displaystyle \sum_{i \mathop \in I} \alpha_i P_i$ be a diagonalizable operator on $H$. Let $B \in \map B H$ be a bounded linear operator. Then the following are equivalent: :$(1): \quad A B = B A$ :$(2): \quad$ For all $i \in I$, $\Rng {P_i}$ is a reducing subspace for $B$ where $\Rng {P_i}$ denotes range.


Composition of Mappings is Composition of Relations

Let $S_1, S_2, S_3$ be sets. Let $f_1: S_1 \to S_2$ and $f_2: S_2 \to S_3$ be mappings such that the domain of $f_2$ is the same set as the codomain of $f_1$. Let $f_2 \circ f_1$ be the composite of $f_1$ and $f_2$. Let $f_1$ and $f_2$ be considered as relations on $S_1 \times S_2$ and $S_2 \times S_3$ respectively. Then $f_2 \circ f_1$ defined as the composition of relations coincides with the definition of $f_2 \circ f_1$ as the composition of mappings.


Null Sequence in Exponential Sequence

Let $\sequence {a_n}_{n \mathop \in \N} \in \C$ be a sequence of complex numbers such that: :$\displaystyle \lim_{n \mathop \to +\infty}a_n = 0$ Then: :$\displaystyle \lim_{n \mathop \to +\infty} \paren {1 + \dfrac {a_n} n}^n = 1$


Identity Mapping is Right Identity

Let $S$ and $T$ be sets. Let $f: S \to T$ be a mapping. Then: :$f \circ I_S = f$ where $I_S$ is the identity mapping on $S$, and $\circ$ signifies composition of mappings.


Diagonal Relation is Left Identity

Let $\RR \subseteq S \times T$ be a relation on $S \times T$. Then: :$\Delta_T \circ \RR = \RR$ where $\Delta_T$ is the diagonal relation on $T$, and $\circ$ signifies composition of relations.


Isomorphism Preserves Associativity

Let $\struct {S, \circ}$ and $\struct {T, *}$ be algebraic structures. Let $\phi: \struct {S, \circ} \to \struct {T, *}$ be an isomorphism. Then $\circ$ is associative {{iff}} $*$ is associative.


Isomorphism Preserves Commutativity

Let $\struct {S, \circ}$ and $\struct {T, *}$ be algebraic structures. Let $\phi: \struct {S, \circ} \to \struct {T, *}$ be an isomorphism. Then $\circ$ is commutative {{iff}} $*$ is commutative.


Isomorphism Preserves Inverses

Let $\left({S, \circ}\right)$ and $\left({T, *}\right)$ be algebraic structures. Let $\phi: \left({S, \circ}\right) \to \left({T, *}\right)$ be an isomorphism. Let $\left({S, \circ}\right)$ have an identity $e_S$. Then $x^{-1}$ is an inverse of $x$ for $\circ$ iff $\phi \left({x^{-1}}\right)$ is an inverse of $\phi \left({x}\right)$ for $*$. That is, iff: : $\phi \left({x^{-1}}\right) = \left({\phi \left({x}\right)}\right)^{-1}$


Isomorphism Preserves Identity

Let $\left({S, \circ}\right)$ and $\left({T, *}\right)$ be algebraic structures. Let $\phi: \left({S, \circ}\right) \to \left({T, *}\right)$ be an isomorphism. Then $\circ$ has an identity $e_S$ {{iff}} $\phi \left({e_S}\right)$ is the identity for $*$.


Non-Zero Natural Numbers under Multiplication form Commutative Semigroup

Let $\N_{>0}$ be the set of natural numbers without zero, that is, $\N_{>0} = \N \setminus \set 0$. The structure $\struct {\N_{>0}, \times}$ forms an infinite commutative semigroup.


Isomorphism Preserves Groups

Let $\left({S, \circ}\right)$ and $\left({T, *}\right)$ be algebraic structures. Let $\phi: \left({S, \circ}\right) \to \left({T, *}\right)$ be an isomorphism. If $\left({S, \circ}\right)$ is a group, then so is $\left({T, *}\right)$.


Isomorphism Preserves Semigroups

Let $\struct {S, \circ}$ and $\struct {T, *}$ be algebraic structures. Let $\phi: S \to T$ be an isomorphism. If $\struct {S, \circ}$ is a semigroup, then so is $\struct {T, *}$.


Left Cancellable iff Left Regular Representation Injective

Let $\struct {S, \circ}$ be an algebraic structure. Then $a \in S$ is left cancellable {{iff}} the left regular representation $\map {\lambda_a} x$ is injective.


Right Cancellable iff Right Regular Representation Injective

Let $\struct {S, \circ}$ be an algebraic structure. Then $a \in S$ is right cancellable {{iff}} the right regular representation $\map {\rho_a} x$ is injective.


Isomorphism Preserves Cancellability

Let $\left({S, \circ}\right)$ and $\left({T, *}\right)$ be algebraic structures. Let $\phi: \left({S, \circ}\right) \to \left({T, *}\right)$ be an isomorphism. Then: : $a \in S$ is cancellable in $\left({S, \circ}\right)$ iff $\phi \left({a}\right) \in T$ is cancellable in $\left({T, *}\right)$.


Isomorphism Preserves Left Cancellability

Let $\struct {S, \circ}$ and $\struct {T, *}$ be algebraic structures. Let $\phi: \struct {S, \circ} \to \struct {T, *}$ be an isomorphism. Then: :$a \in S$ is left cancellable in $\struct {S, \circ}$ {{iff}}: :$\map \phi a \in T$ is left cancellable in $\struct {T, *}$.


Isomorphism Preserves Right Cancellability

Let $\struct {S, \circ}$ and $\struct {T, *}$ be algebraic structures. Let $\phi: \struct {S, \circ} \to \struct {T, *}$ be an isomorphism. Then: :$a \in S$ is right cancellable in $\struct {S, \circ}$ {{iff}} $\map \phi a \in T$ is right cancellable in $\struct {T, *}$.


Differentiation of Power Series

Let $\xi \in \R$ be a real number. Let $\left \langle {a_n} \right \rangle$ be a sequence in $\R$. Let $\displaystyle \sum_{m \mathop \ge 0} a_m \left({x - \xi}\right)^m$ be the power series in $x$ about the point $\xi$. Then within the interval of convergence: :$\displaystyle \frac {\mathrm d^n} {\mathrm d x^n} \sum_{m \mathop \ge 0} a_m \left({x - \xi}\right)^m = \sum_{m \mathop \ge n} a_m m^{\underline n} \left({x - \xi}\right)^{m - n}$ where $m^{\underline n}$ denotes the falling factorial.


General Associativity Theorem/Formulation 1

Let $\struct {S, \circ}$ be a semigroup. Let $\sequence {a_k}_{p + 1 \mathop \le k \mathop \le p + n}$ be a sequence of elements of $S$. Let $\sequence {r_k}_{0 \mathop \le k \mathop \le s}$ be a strictly increasing sequence of natural numbers such that $r_0 = p$ and $r_s = p+n$. Suppose: :$\displaystyle \forall k \in \closedint 1 s: b_k = \prod_{j \mathop = r_{k - 1} \mathop + 1}^{r_k} {a_j}$ Then: :$\displaystyle \forall n \in \N_{>0}: \prod_{k \mathop = 1}^s {b_k} = \prod_{k \mathop = p \mathop + 1}^{p \mathop + n} {a_k}$ That is: :$\displaystyle \forall n \in \N_{>0}: \prod_{k \mathop = 1}^s \paren {a_{r_{k - 1} + 1} \circ a_{r_{k - 1} + 2} \circ \ldots \circ a_{r_k} } = a_{p + 1} \circ \ldots \circ a_{p + n}$


General Associativity Theorem/Formulation 2

Let $n \in \N_{>0}$ and let $a_1, \ldots, a_n$ be elements of a set $S$. Let $\circ$ be an associative operation on $S$. Let the set $\map {P_n} {a_1, a_2, \ldots, a_n}$ be defined inductively by: :$\map {P_1} {a_1} = \set {a_1}$ :$\map {P_2} {a_1, a_2} = \set {a_1 \circ a_2}$ :$\map {P_n} {a_1, a_2, \ldots, a_n} = \set {x \circ y: x \in \map {P_r} {a_1, a_2, \ldots, a_r} \land y \in \map {P_s} {a_{r + 1}, a_{r + 2}, \ldots, a_{r + s} }, n = r + s}$ Then $\map {P_n} {a_1, a_2, \ldots, a_n}$ consists of a unique entity which we can denote $a_1 \circ a_2 \circ \ldots \circ a_n$.


Monoid of Self-Inverse Elements is Abelian Group

Let $\left({S, \circ}\right)$ be a monoid such that: :$\forall x \in S: x \circ x = e$ where $e$ is the identity element of $\left({S, \circ}\right)$. Then $\left({S, \circ}\right)$ is an abelian group.


Product is Right Identity Therefore Right Cancellable

Let $\left({S, \circ}\right)$ be a semigroup. Let $e_R \in S$  be a right identity of $S$. Let $a \in S$ such that: :$\exists b \in S: a \circ b = e_R$ Then $a$ is right cancellable in $\left({S, \circ}\right)$.


Integer Multiples Greater than Positive Integer Closed under Multiplication

Let $n \Z$ be the set of integer multiples of $n$. Let $p \in \Z: p \ge 0$ be a positive integer. Let $S \subseteq n \Z$  be defined as: :$S := \left\{{x \in n \Z: x > p}\right\}$ that is, the set of integer multiples of $n$ greater than $p$. Then the algebraic structure $\struct {S, \times}$ is closed under multiplication.


Non-Zero Complex Numbers Closed under Multiplication

The set of non-zero complex numbers is closed under multiplication.


Non-Zero Modulo Numbers Closed under Multiplication then Modulo is Prime

Let $\left({\Z_m, +_m, \times_m}\right)$ be the ring of integers modulo $m$ for $m > 1$. Let $\Z'_m$ be the set of non-zero integers modulo $m$. Let $\left({\Z_m, \times_m}\right)$ be closed under modulo multiplication. Then $m$ is prime.


Magma is Submagma of Itself

Let $\left({S, \circ}\right)$ be a magma. Then $\left({S, \circ}\right)$ is a submagma of itself.


Empty Set is Submagma of Magma

Let $\left({S, \circ}\right)$ be a magma. Then: : $\left({\varnothing, \circ}\right)$ is a submagma of $\left({S, \circ}\right)$ where $\varnothing$ is the empty set.


Max Operation on Toset is Semigroup

Let $\left({S, \preceq}\right)$ be a totally ordered set. Let $\max \left({x, y}\right)$ denote the max operation on $x, y \in S$. Then $\left({S, \max}\right)$ is a semigroup.


Min Semigroup is Commutative

Let $\left({S, \preceq}\right)$ be a totally ordered set.  Then the semigroup $\left({S, \min}\right)$ is commutative.


Min Semigroup is Idempotent

Let $\struct {S, \preceq}$ be a totally ordered set. Then the semigroup $\struct {S, \min}$ is an idempotent semigroup.


Max Operation on Woset is Monoid

Let $\struct {S, \preceq}$ be a well-ordered set. Let $\max \set {x, y}$ denote the max operation on $x, y \in S$. Then $\struct {S, \max}$ is a monoid. Its identity element is the smallest element of $S$.


Max Operation on Natural Numbers forms Monoid

Let $\left({\N, \max}\right)$ denote the algebraic structure formed from the natural numbers $\N$ and the max operation. Then $\left({\N, \max}\right)$ is a monoid. Its identity element is the zero.


Complex Modulus Function is Continuous

Let $z_0 \in \C$ be a complex number. Then the complex modulus function is continuous at $z_0$.


Euclidean Space is Banach Space

Let $m$ be a positive integer. Then the Euclidean space $\R^m$, along with the Euclidean norm, forms a Banach space over $\R$.


Complex Plane is Complete Metric Space

The complex plane, along with the metric induced by the norm given by the complex modulus, forms a complete metric space.


Euclidean Space is Complete Metric Space

Let $m$ be a positive integer. Then the Euclidean space $\R^m$, along with the Euclidean metric, forms a complete metric space.


Complex Plane is Banach Space

The complex plane, along with the complex modulus, forms a Banach space over $\C$.


Rational Number Space is not Complete Metric Space

Let $\struct {\Q, \tau_d}$ be the rational number space under the Euclidean metric $d$. Then $\struct {\Q, \tau_d}$ is not a complete metric space.


Limit of Subsequence equals Limit of Sequence/Real Numbers

Let $\sequence {x_n}$ be a sequence in $\R$. Let $l \in \R$ such that: :$\displaystyle \lim_{n \mathop \to \infty} x_n = l$ Let $\sequence {x_{n_r} }$ be a subsequence of $\sequence {x_n}$. Then: :$\displaystyle \lim_{r \mathop \to \infty} x_{n_r} = l$


Distance Function of Metric Space is Continuous

Let $M = \struct {A, d}$ be a metric space. Let $\tau_A$ be the topology on $A$ induced by $d$. Let $\struct {A \times A, \tau}$ be the product space of $\struct {A, \tau_A}$ and itself. Then the distance function $d: A \times A \to \R$ is a continuous mapping.


Sequence Converges to Within Half Limit/Complex Numbers

Let $\sequence {z_n}$ be a sequence in $\C$. Let $\sequence {z_n}$ be convergent to the limit $l$. That is, let $\displaystyle \lim_{n \mathop \to \infty} z_n = l$ where $l \ne 0$. Then: : $\exists N: \forall n > N: \cmod {z_n} > \dfrac {\cmod l} 2$


Number to Reciprocal Power is Decreasing

The real sequence $\sequence {n^{1/n} }$ is decreasing for $n \ge 3$.


Inverse of Diagonal Matrix

Let: :$\mathbf D = \begin{bmatrix} a_{11} & 0 & \cdots & 0 \\ 0 & a_{22} & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & a_{nn} \\ \end{bmatrix}$ be an $n \times n$ diagonal matrix. Then its inverse is given by: : $\mathbf D^{-1} = \begin{bmatrix} \dfrac 1 {a_{11}} & 0 & \cdots & 0 \\ 0 & \dfrac 1 {a_{22}} & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & \dfrac 1 {a_{nn}} \\ \end{bmatrix}$ provided that none of the diagonal elements are zero. If any of the diagonal elements are zero, $\mathbf D$ is not invertible.  {{expand|If any diagonal element isn't a unit}}


Subspace of Complete Metric Space is Closed iff Complete

Let $\struct {M, d}$ be a complete metric space. Let $\struct {S, d}$ be a subspace of $\struct {M, d}$. Then $S$ is closed {{iff}} $S$ is complete.


Set of Chains is Chain Complete for Inclusion

Let $\left({P, \leq}\right)$ be an ordered set. Let $\operatorname{ch} \left({P}\right)$ be the set of chains in $P$. Let $\mathcal C$ be a chain in $\operatorname{ch} \left({P}\right)$ with respect to the subset ordering. Then the union $\bigcup \mathcal C$ of $\mathcal C$ is also a chain in $P$.


Jensen's Inequality (Real Analysis)

Let $I$ be a real interval. Let $\phi: I \to \R$ be a convex function. Let $x_1, x_2, \ldots, x_n \in I$. Let $\lambda_1, \lambda_2, \ldots, \lambda_n \ge 0$ be real numbers, at least one of which is non-zero. Then: :$\displaystyle \map \phi {\frac {\sum_{k \mathop = 1}^n \lambda_k x_k} {\sum_{k \mathop = 1}^n \lambda_k} } \le \frac {\sum_{k \mathop = 1}^n \lambda_k \map \phi {x_k} } {\sum_{k \mathop = 1}^n \lambda_k}$ For $\phi$ strictly convex, equality holds {{iff}} $x_1 = x_2 = \cdots = x_n$.


Real Linear Subspace Contains Zero Vector

Let $\mathbb W \subseteq \R^n$ such that $\mathbb W$ is a linear subspace of $\R^n$. Then $\mathbb W$ contains the zero vector: :$\mathbf 0_{n \times 1} = \begin{bmatrix} 0 \\ 0 \\ \vdots \\ 0 \end{bmatrix} \in \mathbb W$


Zero Subspace is Subspace

Let $V$ be a vector space over $K$ with zero vector $\mathbf 0$. The zero subspace $\left\{{\mathbf 0}\right\}$ is a subspace of $V$.


Cartesian Product of Natural Numbers with Itself is Countable

The Cartesian product $\N \times \N$ of the set of natural numbers $\N$ with itself is countable.


Minkowski's Inequality for Sums

Let $a_1, a_2, \ldots, a_n, b_1, b_2, \ldots, b_n \in \R_{\ge 0}$ be non-negative real numbers. Let $p \in \R$, $p \ne 0$ be a real number. If $p < 0$, then we require that $a_1, a_2, \ldots, a_n, b_1, b_2, \ldots, b_n$ be strictly positive. If $p > 1$, then: :$\displaystyle \left({\sum_{k \mathop = 1}^n \left({a_k + b_k}\right)^p}\right)^{1/p} \le \left({\sum_{k \mathop = 1}^n a_k^p}\right)^{1/p} + \left({\sum_{k \mathop = 1}^n b_k^p}\right)^{1/p}$ If $p < 1$, $p \ne 0$, then: :$\displaystyle \left({\sum_{k \mathop = 1}^n \left({a_k + b_k}\right)^p}\right)^{1/p} \ge \left({\sum_{k \mathop = 1}^n a_k^p}\right)^{1/p} + \left({\sum_{k \mathop = 1}^n b_k^p}\right)^{1/p}$


Minkowski's Inequality for Integrals

Let $f, g$ be integrable functions in $X \subseteq \R^n$ with respect to the volume element $dV$. {{explain|Riemann integrable or Lebesgue integrable?}} :$(1):\quad$ Let $p > 1$. Then: :: $\displaystyle \paren {\int_X \size {f + g}^p \rd V}^{1/p} \le \paren {\int_X \size f^p \rd V}^{1/p} + \paren {\int_X \size g^p \rd V}^{1/p}$ :$(2):\quad$ Let $p < 1, p \ne 0$. Then: :: $\displaystyle \paren {\int_X \size {f + g}^p \rd V}^{1/p} \ge \paren {\int_X \size f^p \rd V}^{1/p} + \paren {\int_X \size g^p \rd V}^{1/p}$


Characteristic Function Determined by 1-Fiber

Let $A \subseteq S$. Let $f:S \to \left\{{0, 1}\right\}$ be a mapping. Denote by $\chi_A$ the characteristic function on $A$. Then the following are equivalent: :$(1):\quad f = \chi_A$ :$(2):\quad \forall s \in S: f \left({s}\right) = 1 \iff s \in A$ Using the notion of a fiber, $(2)$ may also be expressed as: :$(2'):\quad f^{-1} \left({1}\right) = A$


Characteristic Function of Intersection

Let $A, B \subseteq S$. Let $\chi_{A \cap B}$ be the characteristic function of their intersection $A \cap B$.


Characteristic Function of Union

Let $A, B \subseteq S$. Let $\chi_{A \mathop \cup B}$ be the characteristic function of their union $A \cup B$.


Characteristic Function of Set Difference

Let $A, B \subseteq S$. Then: :$\chi_{A \mathop \setminus B} = \chi_A - \chi_{A \cap B}$  where: :$A \setminus B$ denotes set difference :$\chi$ denotes characteristic function.


Characteristic Function of Symmetric Difference

Let $A, B \subseteq S$. Then: :$\chi_{A * B} = \chi_A + \chi_B - 2 \chi_{A \cap B}$  where: : $\chi$ denotes characteristic function : $*$ denotes symmetric difference.


Vector Subspace of Real Vector Space

Let $\R^n$ be a real vector space. Let $\mathbb W \subseteq \R^n$. Then $\mathbb W$ is a linear subspace of $\R^n$ {{iff}}: :$(1): \quad \mathbf 0 \in \mathbb W$, where $\mathbf 0$ is the zero vector with $n$ entries :$(2): \quad \mathbb W$ is closed under vector addition :$(3): \quad \mathbb W$ is closed under scalar multiplication.


Reverse Young's Inequality for Products

Let $p, q \in \R_{> 0}$ be strictly positive real numbers satisfying: :$\displaystyle \frac 1 p - \frac 1 q = 1$ Let $a \in \R_{\ge 0}$ be a positive real number. Let $b \in \R_{> 0}$ be a strictly positive real number. Then: :$\displaystyle a b \ge \frac {a^p} p - \frac {b^{-q}} q$


Reverse Hölder's Inequality for Sums

Let $p, q \in \R_{>0}$ be strictly positive real numbers such that $\dfrac 1 p - \dfrac 1 q = 1$. Suppose that the sequences $\mathbf x = \sequence {x_n}$ and $\mathbf y = \sequence {y_n}$ in $\C$ (or $\R$) are such that the series :$\displaystyle \paren {\sum_{n \mathop = 1}^\infty \size {x_n}^p}^{1/p}$ and :$\displaystyle \paren {\sum_{n \mathop = 1}^\infty \size {y_n}^{-q} }^{-1/q}$ are convergent. Denote these values by $\norm {\mathbf x}_p$ and $\norm {\mathbf y}_{-q}$, respectively. Here, the notation $\norm {\mathbf x}_p$ does not denote a norm, but is instead just a convenient notation similar to that of the $p$-norm, which is only defined when $p \ge 1$. Let $\norm {\mathbf {x y} }_1$ denote the $1$-norm of $\mathbf {x y}$, if $\mathbf {x y}$ is in the Lebesgue space $\ell^1$. Then $\norm {\mathbf {x y} }_1 \ge \norm {\mathbf x}_p \norm {\mathbf y}_{-q}$.


Real Vector Space is Vector Space

Let $\R$ be the set of real numbers. Then the real vector space $\R^n$ is a vector space.


Sigma-Algebra Contains Empty Set

Let $X$ be a set, and let $\Sigma$ be a $\sigma$-algebra on $X$. Then $\varnothing \in \Sigma$.


Sigma-Algebra Closed under Union

Let $X$ be a set, and let $\Sigma$ be a $\sigma$-algebra on $X$. Let $A, B \in \Sigma$ be measurable sets. Then $A \cup B \in \Sigma$, where $\cup$ denotes set union.


Sigma-Algebra Closed under Countable Intersection

Let $X$ be a set, and let $\Sigma$ be a $\sigma$-algebra on $X$. Suppose that $\left({E_n}\right)_{n \in \N} \in \Sigma$ is a collection of measurable sets. Then $\displaystyle \bigcap_{n \mathop \in \N} E_n \in \Sigma$, where $\displaystyle \bigcap$ denotes set intersection.


Sigma-Algebra of Countable Sets

Let $X$ be a set. Let $\Sigma$ be the collection of countable and co-countable subsets of $X$. Then $\Sigma$ is a $\sigma$-algebra.


Trace Sigma-Algebra is Sigma-Algebra

Let $X$ be a set, and let $\Sigma$ be a $\sigma$-algebra on $X$. Let $E \subseteq X$ be a subset of $X$. Then the trace $\sigma$-algebra $\Sigma_E$ is a $\sigma$-algebra on $E$.


Pre-Image Sigma-Algebra on Domain is Sigma-Algebra

Let $X, X'$ be sets, and let $f: X \to X'$ be a mapping. Let $\Sigma'$ be a $\sigma$-algebra on $X'$. Then $f^{-1} \left({\Sigma'}\right)$, the pre-image $\sigma$-algebra on the domain of $f$, is a $\sigma$-algebra on $X$.


Intersection of Sigma-Algebras

Let $X$ be a set. Let $\family {\Sigma_i}_{i \mathop \in I}$ be a indexed set of $\sigma$-algebras on $X$. Then $\Sigma := \displaystyle \bigcap_{i \mathop \in I} \Sigma_i$ is also a $\sigma$-algebra on $X$. Here, $\displaystyle \bigcap$ denotes set intersection.


Generated Sigma-Algebra Preserves Subset

Let $X$ be a set. Let $\mathcal F, \mathcal G \subseteq \mathcal P \left({X}\right)$ be collections of subsets of $X$. Suppose that $\mathcal F \subseteq \mathcal G$. Then $\sigma \left({\mathcal F}\right) \subseteq \sigma \left({\mathcal G}\right)$, where $\sigma \left({\mathcal G}\right)$ denotes the $\sigma$-algebra generated by $\mathcal G$


Characterization of Euclidean Borel Sigma-Algebra

Let $\mathcal{O}^n$, $\mathcal{C}^n$ and $\mathcal{K}^n$ be the collections of open, closed and compact subsets of the Euclidean space $\left({\R^n, \tau}\right)$, respectively. Let $\mathcal{J}_{ho}^n$ be the collection of half-open rectangles in $\R^n$. Let $\mathcal{J}^n_{ho, \text{rat}}$ be the collection of half-open rectangles in $\R^n$ with rational endpoints. Then the Borel $\sigma$-algebra $\mathcal B \left({\R^n}\right)$ satisfies: :$\mathcal B \left({\R^n}\right) = \sigma \left({\mathcal{O}^n}\right) = \sigma \left({\mathcal{C}^n}\right) = \sigma \left({\mathcal{K}^n}\right) = \sigma \left({\mathcal{J}_{ho}^n}\right) = \sigma \left({\mathcal{J}^n_{ho, \text{rat}}}\right)$ where $\sigma$ denotes generated $\sigma$-algebra.


Generated Sigma-Algebra Preserves Finiteness

Let $X$ be a set, and let $A_1, \ldots, A_n \subseteq X$ be subsets of $X$. Then $\sigma \left({\left\{{A_1, \ldots, A_n}\right\}}\right)$ is a finite set, where $\sigma$ denotes generated $\sigma$-algebra.


Set of Invertible Mappings forms Symmetric Group

Let $S$ be a set. Let $\struct {S^S, \circ}$ be the monoid of all the mappings from $S$ to itself. Let $\mathcal G$ be the set of all invertible mappings from $S$ to $S$. Then $\struct {\mathcal G, \circ}$ is the symmetric group on $S$.


Group of Units of Field

Let $k$ be a field. Then $k^\times = k \setminus \set 0$


Automorphism Group is Subgroup of Symmetric Group

Let $\struct {S, *}$ be an algebraic structure. Let $\Aut S$ be the automorphism group of $\struct {S, *}$. Then $\Aut S$ is a subgroup of the symmetric group  $\struct {\Gamma \paren S, \circ}$ on $S$.


Differentiation of Vector-Valued Function Componentwise

Let: :$\mathbf r: t \mapsto \left\langle{r_1 \left({t}\right), r_2 \left({t}\right), \ldots, r_n \left({t}\right)}\right\rangle$  be a differentiable vector-valued function. The derivative of a vector-valued function can be calculated by differentiating each of its component functions: :$\dfrac {\d \mathbf r \left({t}\right)} {\d t} = \left\langle{D_t r_1 \left({t}\right), D_t r_2 \left({t}\right), \ldots, D_t r_n \left({t}\right)}\right\rangle$