hcju/proofwikips huggingface dataset:
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  $G$ is a prime group.
Cyclic Group is Simple iff Prime
Let $G$ be a cyclic group. Then $G$ is simple  $G$ 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 : :$(1): \quad H$ 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  $G$ 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 : :$\RR \cap \RR^{-1}$ 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 : :$\mathcal R \cup \mathcal R^{-1} \cup \Delta_S = S \times S$ 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, $a \prec b$  $a$ 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 $T_* \cap T \ne \O$ : :$T$ 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.