hcju/proofwikips huggingface dataset:
Stabilizer of Subspace stabilizes Orthogonal Complement
Let $H$ be a finite-dimensional real or complex Hilbert space (that is, inner product space). Let $t: H \to H$ be a normal operator on $H$. Let $t$ stabilize a subspace $V$. Then $t$ also stabilizes its orthogonal complement $V^\perp$.
Primitive of Secant Function/Secant plus Tangent Form
:$\ds \int \sec x \rd x = \ln \size {\sec x + \tan x} + C$ where $\sec x + \tan x \ne 0$.
Primitive of Cosecant Function/Cosecant plus Cotangent Form
:$\ds \int \csc x \rd x = -\ln \size {\csc x + \cot x} + C$ where $\csc x + \cot x \ne 0$.
Subset of Natural Numbers is Cofinal iff Infinite
Consider the ordered set $\left({\N, \le}\right)$, where $\le$ is the usual ordering on the natural numbers. Let $S \subseteq \N$. Then $S$ is cofinal  it is infinite.
Inner Limit in Hausdorff Space by Set Closures
Let $\struct {\XX, \tau}$ be a Hausdorff space. Let $\sequence {C_n}_{n \mathop \in \N}$ be a sequence of sets in $\XX$. Then: :$\displaystyle \liminf_n C_n = \bigcap_{N \mathop \in \NN_\infty^\#} \map \cl {\bigcup_{n \mathop \in N} C_n}$ where: :$\cl$ denotes set closure :$\NN_\infty^\#$ denotes the set of cofinal subsets of $\N$.
Ordinal is Transitive
Every ordinal (by Definition 3) is a transitive set.
Foundational Relation has no Relational Loops
Let $\prec$ be a foundational relation on $A$ and let $x_1, x_2, \ldots, x_n \in A$. Then: : $\neg \left({x_1 \prec x_2 \land x_3 \prec x_4 \cdots \land x_n \prec x_1}\right)$ That is, there are no relational loops within $A$.
Epsilon is Foundational
Let $\Epsilon$ denote the epsilon relation. Then $\Epsilon$ is a foundational relation on every class $A$.
Inner Limit in Normed Spaces by Open Balls
Let $\left \langle{C_n}\right \rangle_{n \in \N}$ be a sequence of sets in a normed vector space $\left({\mathcal X, \left\Vert{\cdot}\right\Vert}\right)$. Then the inner limit of $\left \langle{C_n}\right \rangle_{n \in \N}$ is: : $\displaystyle \liminf_n \ C_n = \left\{{x: \forall \epsilon > 0: \exists N \in \mathcal N_\infty: \forall n \in N: x \in C_n + \epsilon B}\right\}$ where $B$ denotes the open unit ball of the space. {{explain|What are $N$ and $\mathcal N_\infty$ in this context? Also, what point is at the center of $B$? And what does $C_n + \epsilon B$ mean? For the latter one suspects $\cup$, but this needs to be checked. This page might need to be rewritten from a new perspective, as the original author was touchy about symbols used and departed ProofWiki in a rage when his notation was changed.}}
Inner Limit is Closed Set
Let $\struct {S, \tau}$ be a Hausdorff topological space. Let $\sequence {C_n}_{n \mathop \in \N}$ be a sequence of sets in $S$. Then the inner limit $\liminf_n C_n$ is a closed set.
Local Basis of Topological Vector Space
Let $\struct {\XX, \tau}$ be a topological vector space. Let $0_\XX$ denote the zero vector of $\XX$. Then there exists a local basis $\BB$ of $0_\XX$ with the following properties: :$(1): \quad \forall W \in \BB: \exists V \in \BB$ such that $V + V \subseteq W$ (where the addition $V + V$ is meant in the sense of the Minkowski sum) :$(2): \quad$ Every $W \in \BB$ is star-shaped (balanced) :$(3): \quad$ Every $W \in \BB$ is absorbent. :$(4): \quad \displaystyle \bigcap_\BB W = \set {0_\XX}$.
No Membership Loops
For any proper classes or sets $A_1, A_2, \ldots, A_n$: :$\neg \paren {A_1 \in A_2 \land A_2 \in A_3 \land \cdots \land A_n \in A_1}$
Composite of Isomorphisms is Isomorphism/Algebraic Structure
Let: : $\left({S_1, \circ_1, \circ_2, \ldots, \circ_n}\right)$ : $\left({S_2, *_1, *_2, \ldots, *_n}\right)$ : $\left({S_3, \oplus_1, \oplus_2, \ldots, \oplus_n}\right)$ be algebraic structures. Let: : $\phi: \left({S_1, \circ_1, \circ_2, \ldots, \circ_n}\right) \to \left({S_2, *_1, *_2, \ldots, *_n}\right)$ : $\psi: \left({S_2, *_1, *_2, \ldots, *_n}\right) \to \left({S_3, \oplus_1, \oplus_2, \ldots, \oplus_n}\right)$ be isomorphisms. Then the composite of $\phi$ and $\psi$ is also an isomorphism.
Composite of Isomorphisms is Isomorphism/R-Algebraic Structure
Let: : $\left({S_1, \ast_1}\right)_R$ : $\left({S_2, \ast_2}\right)_R$ : $\left({S_3, \ast_3}\right)_R$ be $R$-algebraic structures with the same number of operations. Let: : $\phi: \left({S_1, \ast_1}\right)_R \to \left({S_2, \ast_2}\right)_R$ : $\psi: \left({S_2, \ast_2}\right)_R \to \left({S_3, \ast_3}\right)_R$ be isomorphisms. Then the composite of $\phi$ and $\psi$ is also an isomorphism.
Element of Transitive Class
Let $B$ be a transitive class. Then: :$A \in B \implies A \subsetneq B$ where $\subsetneq$ denotes a proper subset).
Restriction of Foundational Relation is Foundational
Let $S$ be a set. Let $\RR \subseteq S \times S$ be a foundational relation on $S$. Let $T \subseteq S$ be a subset of $S$. Let $\RR {\restriction_T} \subseteq T \times T$ be the restriction of $\RR$ to $T$. Then $\RR {\restriction_T}$ is a foundational relation on $T$.
Restriction of Strict Well-Ordering is Strict Well-Ordering
Let $R$ be a strict well-ordering of $A$. Let $B \subseteq A$. Then $R$ is a strict well-ordering of $B$.
Alternative Definition of Ordinal
A set $S$ is an ordinal  $S$ is transitive and is strictly well-ordered by the $\in$-relation.
Ordinal Class is Ordinal
The Ordinal Class $\On$ is an ordinal.
Transitive Set is Proper Subset of Ordinal iff Element of Ordinal
Let $A$ be an ordinal. Let $B$ be a transitive set. Then: :$B \subsetneq A \iff B \in A$
Ordinal Membership is Trichotomy
Let $A$ and $B$ be ordinals. Then: :$\left({A = B}\right) \lor \left({A \in B}\right) \lor \left({B \in A}\right)$ where $\lor$ denotes logical or.
Second Derivative of Concave Real Function is Non-Positive
Let $f$ be a real function which is twice differentiable on the open interval $\openint a b$. Then $f$ is concave on $\openint a b$  its second derivative $f'' \le 0$ on $\openint a b$.
Ordinal is Member of Ordinal Class
Let $A$ be an ordinal. Then: :$A \in \On \lor A = \On$ where $\On$ denote the class of ordinals .
De Morgan's Laws (Set Theory)/Set Difference
{{:De Morgan's Laws (Set Theory)/Set Difference/Difference with Intersection}}
De Morgan's Laws (Set Theory)/Set Difference/General Case
Let $S$ and $T$ be sets. Let $\mathcal P \left({T}\right)$ be the power set of $T$. Let $\mathbb T \subseteq \mathcal P \left({T}\right)$. Then: ==== Difference with Intersection ==== {{:De Morgan's Laws (Set Theory)/Set Difference/General Case/Difference with Intersection}} ==== Difference with Union ==== {{:De Morgan's Laws (Set Theory)/Set Difference/General Case/Difference with Union}}
De Morgan's Laws (Set Theory)/Set Complement
{{:De Morgan's Laws (Set Theory)/Set Complement/Complement of Intersection}}
Complement of Preimage equals Preimage of Complement
Let $f: S \to T$ be a mapping. Let $T_1$ be a subset of $T$. Then: :$\relcomp S {f^{-1} \sqbrk {T_1} } = f^{-1} \sqbrk {\relcomp T {T_1} }$ where: :$\complement_S$ (in this context) denotes relative complement :$f^{-1} \sqbrk {T_1}$ denotes preimage.
One-to-Many Image of Set Difference/Corollary 1
Let $\RR \subseteq S \times T$ be a relation which is one-to-many. Let $A \subseteq B \subseteq S$. Then: :$\relcomp {\RR \sqbrk B} {\RR \sqbrk A} = \RR \sqbrk {\relcomp B A}$ where $\complement$ (in this context) denotes relative complement.
One-to-Many Image of Set Difference/Corollary 2
Let $\mathcal R \subseteq S \times T$ be a relation which is one-to-many. Let $A$ be a subset of $S$. Then: :$\relcomp {\Img {\mathcal R} } {\mathcal R \sqbrk A} = \mathcal R \sqbrk {\relcomp S A}$ where $\complement$ (in this context) denotes relative complement. In the language of direct image mappings this can be presented as: :$\forall A \in \powerset S: \map {\paren {\complement_{\Img {\mathcal R} } \circ \mathcal R^\to} } A = \map {\paren {\mathcal R^\to \circ \complement_S} } A$
Integer Power Function is Bijective iff Index is Odd
Let $n \in \Z_{\ge 0}$ be a positive integer. Let $f_n: \R \to \R$ be the real function defined as: :$\map {f_n} x = x^n$ Then $f_n$ is a bijection  $n$ is odd.
Set of Infinite Sequences is Uncountable
Let $S$ be a set which contains more than one element. Let $S^\infty$ denote the set of all sequences of elements of $S$. Then $S^\infty$ is uncountable.
Union of Topologies is not necessarily Topology
Let $\left({\tau_i}\right)_{i \in I}$ be an arbitrary indexed set of topologies for a set $S$. Then $\tau := \displaystyle \bigcup_{i \mathop \in I} {\tau_i}$ is not necessarily also a topology for $S$.
Niven's Theorem
Consider the angles $\theta$ in the range $0 \le \theta \le \dfrac \pi 2$. The only values of $\theta$ such that both $\dfrac \theta \pi$ and $\sin \theta$ are rational are: :$\theta = 0: \sin \theta = 0$ :$\theta = \dfrac \pi 6: \sin \theta = \dfrac 1 2$ :$\theta = \dfrac \pi 2: \sin \theta = 1$
Sorgenfrey Line is Topology
The Sorgenfrey Line is a topological space.
Sorgenfrey Line is Hausdorff
Let $T = \struct {\R, \tau}$ be the Sorgenfrey line. Then $T$ is Hausdorff.
Sorgenfrey Line is Perfectly Normal
Let $T = \struct {\R, \tau}$ be the Sorgenfrey line. Then $T$ is perfectly normal.
Sorgenfrey Line is Expansion of Real Line
Let $\R = \struct {\R, d}$ be the metric space defined in Real Number Line is Metric Space. Let $T = \struct {\R, \tau}$ be the Sorgenfrey line. Then $T$ is an expansion of $\R$ as a topological space.
Sorgenfrey Line satisfies all Separation Axioms
Let $T = \left({\R, \tau}\right)$ be the Sorgenfrey line. Then $T$ satisfies all separation axioms.
Argument of Product equals Sum of Arguments
Let $z_1, z_2 \in \C$ be complex numbers. Let $\arg$ be the argument operator. Then: :$\map \arg {z_1 z_2} = \map \arg {z_1} + \map \arg {z_2} + 2 k \pi$ where $k$ can be $0$, $1$ or $-1$.
Convergence of Sequence in Discrete Space
Let $T = \left({S, \tau}\right)$ be a discrete topological space. Let $H = \left \langle{x_n}\right \rangle_{n \in \N}$ be a sequence in $S$. Then $H$ converges in $T$ to a limit iff: :$\exists k \in \N: \forall m \in \N: m > k: x_m = x_k$ That is, iff the sequence reaches some value of $S$ and "stays there".
Discrete Subspace of Fortissimo Space
Let $T = \left({S, \tau_p}\right)$ be a Fortissimo space. Let $T' = \left({S \setminus \left\{{p}\right\}, \tau_p}\right)$ be the topological subspace induced on $T$ by the subset $S \setminus \left\{{p}\right\}$. Then $T'$ is a discrete topological space.
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 $T$ : :$p \in H$ 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  $A = \set {\tuple {0, 0} }$.
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  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 : :$\exists k \in \Z: \forall n \in \Z: \map \phi n = k n$
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 $n$  $\struct {G, \circ}$ 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<br/><br/>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  $K^\perp = \left({0}\right)$, 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 $G$ : :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