In functional analysis and related areas of mathematics, a metrizable (resp. pseudometrizable) topological vector space (TVS) is a TVS whose topology is induced by a metric (resp. pseudometric). An LM-space is an inductive limit of a sequence of locally convex metrizable TVS.
A pseudometric on a set X {\displaystyle X} is a map d : X × × --> X → → --> R {\displaystyle d:X\times X\rightarrow \mathbb {R} } satisfying the following properties:
A pseudometric is called a metric if it satisfies:
Ultrapseudometric
A pseudometric d {\displaystyle d} on X {\displaystyle X} is called a ultrapseudometric or a strong pseudometric if it satisfies:
Pseudometric space
A pseudometric space is a pair ( X , d ) {\displaystyle (X,d)} consisting of a set X {\displaystyle X} and a pseudometric d {\displaystyle d} on X {\displaystyle X} such that X {\displaystyle X} 's topology is identical to the topology on X {\displaystyle X} induced by d . {\displaystyle d.} We call a pseudometric space ( X , d ) {\displaystyle (X,d)} a metric space (resp. ultrapseudometric space) when d {\displaystyle d} is a metric (resp. ultrapseudometric).
If d {\displaystyle d} is a pseudometric on a set X {\displaystyle X} then collection of open balls: B r ( z ) := { x ∈ ∈ --> X : d ( x , z ) < r } {\displaystyle B_{r}(z):=\{x\in X:d(x,z)<r\}} as z {\displaystyle z} ranges over X {\displaystyle X} and r > 0 {\displaystyle r>0} ranges over the positive real numbers, forms a basis for a topology on X {\displaystyle X} that is called the d {\displaystyle d} -topology or the pseudometric topology on X {\displaystyle X} induced by d . {\displaystyle d.}
Pseudometrizable space
A topological space ( X , τ τ --> ) {\displaystyle (X,\tau )} is called pseudometrizable (resp. metrizable, ultrapseudometrizable) if there exists a pseudometric (resp. metric, ultrapseudometric) d {\displaystyle d} on X {\displaystyle X} such that τ τ --> {\displaystyle \tau } is equal to the topology induced by d . {\displaystyle d.} [1]
An additive topological group is an additive group endowed with a topology, called a group topology, under which addition and negation become continuous operators.
A topology τ τ --> {\displaystyle \tau } on a real or complex vector space X {\displaystyle X} is called a vector topology or a TVS topology if it makes the operations of vector addition and scalar multiplication continuous (that is, if it makes X {\displaystyle X} into a topological vector space).
Every topological vector space (TVS) X {\displaystyle X} is an additive commutative topological group but not all group topologies on X {\displaystyle X} are vector topologies. This is because despite it making addition and negation continuous, a group topology on a vector space X {\displaystyle X} may fail to make scalar multiplication continuous. For instance, the discrete topology on any non-trivial vector space makes addition and negation continuous but do not make scalar multiplication continuous.
If X {\displaystyle X} is an additive group then we say that a pseudometric d {\displaystyle d} on X {\displaystyle X} is translation invariant or just invariant if it satisfies any of the following equivalent conditions:
If X {\displaystyle X} is a topological group the a value or G-seminorm on X {\displaystyle X} (the G stands for Group) is a real-valued map p : X → → --> R {\displaystyle p:X\rightarrow \mathbb {R} } with the following properties:[2]
where we call a G-seminorm a G-norm if it satisfies the additional condition:
If p {\displaystyle p} is a value on a vector space X {\displaystyle X} then:
Theorem[2] — Suppose that X {\displaystyle X} is an additive commutative group. If d {\displaystyle d} is a translation invariant pseudometric on X {\displaystyle X} then the map p ( x ) := d ( x , 0 ) {\displaystyle p(x):=d(x,0)} is a value on X {\displaystyle X} called the value associated with d {\displaystyle d} , and moreover, d {\displaystyle d} generates a group topology on X {\displaystyle X} (i.e. the d {\displaystyle d} -topology on X {\displaystyle X} makes X {\displaystyle X} into a topological group). Conversely, if p {\displaystyle p} is a value on X {\displaystyle X} then the map d ( x , y ) := p ( x − − --> y ) {\displaystyle d(x,y):=p(x-y)} is a translation-invariant pseudometric on X {\displaystyle X} and the value associated with d {\displaystyle d} is just p . {\displaystyle p.}
Theorem[2] — If ( X , τ τ --> ) {\displaystyle (X,\tau )} is an additive commutative topological group then the following are equivalent:
If ( X , τ τ --> ) {\displaystyle (X,\tau )} is Hausdorff then the word "pseudometric" in the above statement may be replaced by the word "metric." A commutative topological group is metrizable if and only if it is Hausdorff and pseudometrizable.
Let X {\displaystyle X} be a non-trivial (i.e. X ≠ ≠ --> { 0 } {\displaystyle X\neq \{0\}} ) real or complex vector space and let d {\displaystyle d} be the translation-invariant trivial metric on X {\displaystyle X} defined by d ( x , x ) = 0 {\displaystyle d(x,x)=0} and d ( x , y ) = 1 for all x , y ∈ ∈ --> X {\displaystyle d(x,y)=1{\text{ for all }}x,y\in X} such that x ≠ ≠ --> y . {\displaystyle x\neq y.} The topology τ τ --> {\displaystyle \tau } that d {\displaystyle d} induces on X {\displaystyle X} is the discrete topology, which makes ( X , τ τ --> ) {\displaystyle (X,\tau )} into a commutative topological group under addition but does not form a vector topology on X {\displaystyle X} because ( X , τ τ --> ) {\displaystyle (X,\tau )} is disconnected but every vector topology is connected. What fails is that scalar multiplication isn't continuous on ( X , τ τ --> ) . {\displaystyle (X,\tau ).}
This example shows that a translation-invariant (pseudo)metric is not enough to guarantee a vector topology, which leads us to define paranorms and F-seminorms.
A collection N {\displaystyle {\mathcal {N}}} of subsets of a vector space is called additive[5] if for every N ∈ ∈ --> N , {\displaystyle N\in {\mathcal {N}},} there exists some U ∈ ∈ --> N {\displaystyle U\in {\mathcal {N}}} such that U + U ⊆ ⊆ --> N . {\displaystyle U+U\subseteq N.}
Continuity of addition at 0 — If ( X , + ) {\displaystyle (X,+)} is a group (as all vector spaces are), τ τ --> {\displaystyle \tau } is a topology on X , {\displaystyle X,} and X × × --> X {\displaystyle X\times X} is endowed with the product topology, then the addition map X × × --> X → → --> X {\displaystyle X\times X\to X} (i.e. the map ( x , y ) ↦ ↦ --> x + y {\displaystyle (x,y)\mapsto x+y} ) is continuous at the origin of X × × --> X {\displaystyle X\times X} if and only if the set of neighborhoods of the origin in ( X , τ τ --> ) {\displaystyle (X,\tau )} is additive. This statement remains true if the word "neighborhood" is replaced by "open neighborhood."[5]
All of the above conditions are consequently a necessary for a topology to form a vector topology. Additive sequences of sets have the particularly nice property that they define non-negative continuous real-valued subadditive functions. These functions can then be used to prove many of the basic properties of topological vector spaces and also show that a Hausdorff TVS with a countable basis of neighborhoods is metrizable. The following theorem is true more generally for commutative additive topological groups.
Theorem — Let U ∙ ∙ --> = ( U i ) i = 0 ∞ ∞ --> {\displaystyle U_{\bullet }=\left(U_{i}\right)_{i=0}^{\infty }} be a collection of subsets of a vector space such that 0 ∈ ∈ --> U i {\displaystyle 0\in U_{i}} and U i + 1 + U i + 1 ⊆ ⊆ --> U i {\displaystyle U_{i+1}+U_{i+1}\subseteq U_{i}} for all i ≥ ≥ --> 0. {\displaystyle i\geq 0.} For all u ∈ ∈ --> U 0 , {\displaystyle u\in U_{0},} let S ( u ) := { n ∙ ∙ --> = ( n 1 , … … --> , n k ) : k ≥ ≥ --> 1 , n i ≥ ≥ --> 0 for all i , and u ∈ ∈ --> U n 1 + ⋯ ⋯ --> + U n k } . {\displaystyle \mathbb {S} (u):=\left\{n_{\bullet }=\left(n_{1},\ldots ,n_{k}\right)~:~k\geq 1,n_{i}\geq 0{\text{ for all }}i,{\text{ and }}u\in U_{n_{1}}+\cdots +U_{n_{k}}\right\}.}
Define f : X → → --> [ 0 , 1 ] {\displaystyle f:X\to [0,1]} by f ( x ) = 1 {\displaystyle f(x)=1} if x ∉ U 0 {\displaystyle x\not \in U_{0}} and otherwise let f ( x ) := inf { 2 − − --> n 1 + ⋯ ⋯ --> 2 − − --> n k : n ∙ ∙ --> = ( n 1 , … … --> , n k ) ∈ ∈ --> S ( x ) } . {\displaystyle f(x):=\inf _{}\left\{2^{-n_{1}}+\cdots 2^{-n_{k}}~:~n_{\bullet }=\left(n_{1},\ldots ,n_{k}\right)\in \mathbb {S} (x)\right\}.}
Then f {\displaystyle f} is subadditive (meaning f ( x + y ) ≤ ≤ --> f ( x ) + f ( y ) for all x , y ∈ ∈ --> X {\displaystyle f(x+y)\leq f(x)+f(y){\text{ for all }}x,y\in X} ) and f = 0 {\displaystyle f=0} on ⋂ ⋂ --> i ≥ ≥ --> 0 U i , {\displaystyle \bigcap _{i\geq 0}U_{i},} so in particular f ( 0 ) = 0. {\displaystyle f(0)=0.} If all U i {\displaystyle U_{i}} are symmetric sets then f ( − − --> x ) = f ( x ) {\displaystyle f(-x)=f(x)} and if all U i {\displaystyle U_{i}} are balanced then f ( s x ) ≤ ≤ --> f ( x ) {\displaystyle f(sx)\leq f(x)} for all scalars s {\displaystyle s} such that | s | ≤ ≤ --> 1 {\displaystyle |s|\leq 1} and all x ∈ ∈ --> X . {\displaystyle x\in X.} If X {\displaystyle X} is a topological vector space and if all U i {\displaystyle U_{i}} are neighborhoods of the origin then f {\displaystyle f} is continuous, where if in addition X {\displaystyle X} is Hausdorff and U ∙ ∙ --> {\displaystyle U_{\bullet }} forms a basis of balanced neighborhoods of the origin in X {\displaystyle X} then d ( x , y ) := f ( x − − --> y ) {\displaystyle d(x,y):=f(x-y)} is a metric defining the vector topology on X . {\displaystyle X.}
Assume that n ∙ ∙ --> = ( n 1 , … … --> , n k ) {\displaystyle n_{\bullet }=\left(n_{1},\ldots ,n_{k}\right)} always denotes a finite sequence of non-negative integers and use the notation: ∑ ∑ --> 2 − − --> n ∙ ∙ --> := 2 − − --> n 1 + ⋯ ⋯ --> + 2 − − --> n k and ∑ ∑ --> U n ∙ ∙ --> := U n 1 + ⋯ ⋯ --> + U n k . {\displaystyle \sum 2^{-n_{\bullet }}:=2^{-n_{1}}+\cdots +2^{-n_{k}}\quad {\text{ and }}\quad \sum U_{n_{\bullet }}:=U_{n_{1}}+\cdots +U_{n_{k}}.}
For any integers n ≥ ≥ --> 0 {\displaystyle n\geq 0} and d > 2 , {\displaystyle d>2,} U n ⊇ ⊇ --> U n + 1 + U n + 1 ⊇ ⊇ --> U n + 1 + U n + 2 + U n + 2 ⊇ ⊇ --> U n + 1 + U n + 2 + ⋯ ⋯ --> + U n + d + U n + d + 1 + U n + d + 1 . {\displaystyle U_{n}\supseteq U_{n+1}+U_{n+1}\supseteq U_{n+1}+U_{n+2}+U_{n+2}\supseteq U_{n+1}+U_{n+2}+\cdots +U_{n+d}+U_{n+d+1}+U_{n+d+1}.}
From this it follows that if n ∙ ∙ --> = ( n 1 , … … --> , n k ) {\displaystyle n_{\bullet }=\left(n_{1},\ldots ,n_{k}\right)} consists of distinct positive integers then ∑ ∑ --> U n ∙ ∙ --> ⊆ ⊆ --> U − − --> 1 + min ( n ∙ ∙ --> ) . {\displaystyle \sum U_{n_{\bullet }}\subseteq U_{-1+\min \left(n_{\bullet }\right)}.}
It will now be shown by induction on k {\displaystyle k} that if n ∙ ∙ --> = ( n 1 , … … --> , n k ) {\displaystyle n_{\bullet }=\left(n_{1},\ldots ,n_{k}\right)} consists of non-negative integers such that ∑ ∑ --> 2 − − --> n ∙ ∙ --> ≤ ≤ --> 2 − − --> M {\displaystyle \sum 2^{-n_{\bullet }}\leq 2^{-M}} for some integer M ≥ ≥ --> 0 {\displaystyle M\geq 0} then ∑ ∑ --> U n ∙ ∙ --> ⊆ ⊆ --> U M . {\displaystyle \sum U_{n_{\bullet }}\subseteq U_{M}.} This is clearly true for k = 1 {\displaystyle k=1} and k = 2 {\displaystyle k=2} so assume that k > 2 , {\displaystyle k>2,} which implies that all n i {\displaystyle n_{i}} are positive. If all n i {\displaystyle n_{i}} are distinct then this step is done, and otherwise pick distinct indices i < j {\displaystyle i<j} such that n i = n j {\displaystyle n_{i}=n_{j}} and construct m ∙ ∙ --> = ( m 1 , … … --> , m k − − --> 1 ) {\displaystyle m_{\bullet }=\left(m_{1},\ldots ,m_{k-1}\right)} from n ∙ ∙ --> {\displaystyle n_{\bullet }} by replacing each n i {\displaystyle n_{i}} with n i − − --> 1 {\displaystyle n_{i}-1} and deleting the j th {\displaystyle j^{\text{th}}} element of n ∙ ∙ --> {\displaystyle n_{\bullet }} (all other elements of n ∙ ∙ --> {\displaystyle n_{\bullet }} are transferred to m ∙ ∙ --> {\displaystyle m_{\bullet }} unchanged). Observe that ∑ ∑ --> 2 − − --> n ∙ ∙ --> = ∑ ∑ --> 2 − − --> m ∙ ∙ --> {\displaystyle \sum 2^{-n_{\bullet }}=\sum 2^{-m_{\bullet }}} and ∑ ∑ --> U n ∙ ∙ --> ⊆ ⊆ --> ∑ ∑ --> U m ∙ ∙ --> {\displaystyle \sum U_{n_{\bullet }}\subseteq \sum U_{m_{\bullet }}} (because U n i + U n j ⊆ ⊆ --> U n i − − --> 1 {\displaystyle U_{n_{i}}+U_{n_{j}}\subseteq U_{n_{i}-1}} ) so by appealing to the inductive hypothesis we conclude that ∑ ∑ --> U n ∙ ∙ --> ⊆ ⊆ --> ∑ ∑ --> U m ∙ ∙ --> ⊆ ⊆ --> U M , {\displaystyle \sum U_{n_{\bullet }}\subseteq \sum U_{m_{\bullet }}\subseteq U_{M},} as desired.
It is clear that f ( 0 ) = 0 {\displaystyle f(0)=0} and that 0 ≤ ≤ --> f ≤ ≤ --> 1 {\displaystyle 0\leq f\leq 1} so to prove that f {\displaystyle f} is subadditive, it suffices to prove that f ( x + y ) ≤ ≤ --> f ( x ) + f ( y ) {\displaystyle f(x+y)\leq f(x)+f(y)} when x , y ∈ ∈ --> X {\displaystyle x,y\in X} are such that f ( x ) + f ( y ) < 1 , {\displaystyle f(x)+f(y)<1,} which implies that x , y ∈ ∈ --> U 0 . {\displaystyle x,y\in U_{0}.} This is an exercise. If all U i {\displaystyle U_{i}} are symmetric then x ∈ ∈ --> ∑ ∑ --> U n ∙ ∙ --> {\displaystyle x\in \sum U_{n_{\bullet }}} if and only if − − --> x ∈ ∈ --> ∑ ∑ --> U n ∙ ∙ --> {\displaystyle -x\in \sum U_{n_{\bullet }}} from which it follows that f ( − − --> x ) ≤ ≤ --> f ( x ) {\displaystyle f(-x)\leq f(x)} and f ( − − --> x ) ≥ ≥ --> f ( x ) . {\displaystyle f(-x)\geq f(x).} If all U i {\displaystyle U_{i}} are balanced then the inequality f ( s x ) ≤ ≤ --> f ( x ) {\displaystyle f(sx)\leq f(x)} for all unit scalars s {\displaystyle s} such that | s | ≤ ≤ --> 1 {\displaystyle |s|\leq 1} is proved similarly. Because f {\displaystyle f} is a nonnegative subadditive function satisfying f ( 0 ) = 0 , {\displaystyle f(0)=0,} as described in the article on sublinear functionals, f {\displaystyle f} is uniformly continuous on X {\displaystyle X} if and only if f {\displaystyle f} is continuous at the origin. If all U i {\displaystyle U_{i}} are neighborhoods of the origin then for any real r > 0 , {\displaystyle r>0,} pick an integer M > 1 {\displaystyle M>1} such that 2 − − --> M < r {\displaystyle 2^{-M}<r} so that x ∈ ∈ --> U M {\displaystyle x\in U_{M}} implies f ( x ) ≤ ≤ --> 2 − − --> M < r . {\displaystyle f(x)\leq 2^{-M}<r.} If the set of all U i {\displaystyle U_{i}} form basis of balanced neighborhoods of the origin then it may be shown that for any n > 1 , {\displaystyle n>1,} there exists some 0 < r ≤ ≤ --> 2 − − --> n {\displaystyle 0<r\leq 2^{-n}} such that f ( x ) < r {\displaystyle f(x)<r} implies x ∈ ∈ --> U n . {\displaystyle x\in U_{n}.} ◼ ◼ --> {\displaystyle \blacksquare }
If X {\displaystyle X} is a vector space over the real or complex numbers then a paranorm on X {\displaystyle X} is a G-seminorm (defined above) p : X → → --> R {\displaystyle p:X\rightarrow \mathbb {R} } on X {\displaystyle X} that satisfies any of the following additional conditions, each of which begins with "for all sequences x ∙ ∙ --> = ( x i ) i = 1 ∞ ∞ --> {\displaystyle x_{\bullet }=\left(x_{i}\right)_{i=1}^{\infty }} in X {\displaystyle X} and all convergent sequences of scalars s ∙ ∙ --> = ( s i ) i = 1 ∞ ∞ --> {\displaystyle s_{\bullet }=\left(s_{i}\right)_{i=1}^{\infty }} ":[6]
A paranorm is called total if in addition it satisfies:
If p {\displaystyle p} is a paranorm on a vector space X {\displaystyle X} then the map d : X × × --> X → → --> R {\displaystyle d:X\times X\rightarrow \mathbb {R} } defined by d ( x , y ) := p ( x − − --> y ) {\displaystyle d(x,y):=p(x-y)} is a translation-invariant pseudometric on X {\displaystyle X} that defines a vector topology on X . {\displaystyle X.} [8]
If p {\displaystyle p} is a paranorm on a vector space X {\displaystyle X} then:
If X {\displaystyle X} is a vector space over the real or complex numbers then an F-seminorm on X {\displaystyle X} (the F {\displaystyle F} stands for Fréchet) is a real-valued map p : X → → --> R {\displaystyle p:X\to \mathbb {R} } with the following four properties: [11]
An F-seminorm is called an F-norm if in addition it satisfies:
An F-seminorm is called monotone if it satisfies:
An F-seminormed space (resp. F-normed space)[12] is a pair ( X , p ) {\displaystyle (X,p)} consisting of a vector space X {\displaystyle X} and an F-seminorm (resp. F-norm) p {\displaystyle p} on X . {\displaystyle X.}
If ( X , p ) {\displaystyle (X,p)} and ( Z , q ) {\displaystyle (Z,q)} are F-seminormed spaces then a map f : X → → --> Z {\displaystyle f:X\to Z} is called an isometric embedding[12] if q ( f ( x ) − − --> f ( y ) ) = p ( x , y ) for all x , y ∈ ∈ --> X . {\displaystyle q(f(x)-f(y))=p(x,y){\text{ for all }}x,y\in X.}
Every isometric embedding of one F-seminormed space into another is a topological embedding, but the converse is not true in general.[12]
Every F-seminorm is a paranorm and every paranorm is equivalent to some F-seminorm.[7] Every F-seminorm on a vector space X {\displaystyle X} is a value on X . {\displaystyle X.} In particular, p ( x ) = 0 , {\displaystyle p(x)=0,} and p ( x ) = p ( − − --> x ) {\displaystyle p(x)=p(-x)} for all x ∈ ∈ --> X . {\displaystyle x\in X.}
Theorem[11] — Let p {\displaystyle p} be an F-seminorm on a vector space X . {\displaystyle X.} Then the map d : X × × --> X → → --> R {\displaystyle d:X\times X\to \mathbb {R} } defined by d ( x , y ) := p ( x − − --> y ) {\displaystyle d(x,y):=p(x-y)} is a translation invariant pseudometric on X {\displaystyle X} that defines a vector topology τ τ --> {\displaystyle \tau } on X . {\displaystyle X.} If p {\displaystyle p} is an F-norm then d {\displaystyle d} is a metric. When X {\displaystyle X} is endowed with this topology then p {\displaystyle p} is a continuous map on X . {\displaystyle X.}
The balanced sets { x ∈ ∈ --> X : p ( x ) ≤ ≤ --> r } , {\displaystyle \{x\in X~:~p(x)\leq r\},} as r {\displaystyle r} ranges over the positive reals, form a neighborhood basis at the origin for this topology consisting of closed set. Similarly, the balanced sets { x ∈ ∈ --> X : p ( x ) < r } , {\displaystyle \{x\in X~:~p(x)<r\},} as r {\displaystyle r} ranges over the positive reals, form a neighborhood basis at the origin for this topology consisting of open sets.
Suppose that L {\displaystyle {\mathcal {L}}} is a non-empty collection of F-seminorms on a vector space X {\displaystyle X} and for any finite subset F ⊆ ⊆ --> L {\displaystyle {\mathcal {F}}\subseteq {\mathcal {L}}} and any r > 0 , {\displaystyle r>0,} let U F , r := ⋂ ⋂ --> p ∈ ∈ --> F { x ∈ ∈ --> X : p ( x ) < r } . {\displaystyle U_{{\mathcal {F}},r}:=\bigcap _{p\in {\mathcal {F}}}\{x\in X:p(x)<r\}.}
The set { U F , r : r > 0 , F ⊆ ⊆ --> L , F finite } {\displaystyle \left\{U_{{\mathcal {F}},r}~:~r>0,{\mathcal {F}}\subseteq {\mathcal {L}},{\mathcal {F}}{\text{ finite }}\right\}} forms a filter base on X {\displaystyle X} that also forms a neighborhood basis at the origin for a vector topology on X {\displaystyle X} denoted by τ τ --> L . {\displaystyle \tau _{\mathcal {L}}.} [12] Each U F , r {\displaystyle U_{{\mathcal {F}},r}} is a balanced and absorbing subset of X . {\displaystyle X.} [12] These sets satisfy[12] U F , r / 2 + U F , r / 2 ⊆ ⊆ --> U F , r . {\displaystyle U_{{\mathcal {F}},r/2}+U_{{\mathcal {F}},r/2}\subseteq U_{{\mathcal {F}},r}.}
Suppose that p ∙ ∙ --> = ( p i ) i = 1 ∞ ∞ --> {\displaystyle p_{\bullet }=\left(p_{i}\right)_{i=1}^{\infty }} is a family of non-negative subadditive functions on a vector space X . {\displaystyle X.}
The Fréchet combination[8] of p ∙ ∙ --> {\displaystyle p_{\bullet }} is defined to be the real-valued map p ( x ) := ∑ ∑ --> i = 1 ∞ ∞ --> p i ( x ) 2 i [ 1 + p i ( x ) ] . {\displaystyle p(x):=\sum _{i=1}^{\infty }{\frac {p_{i}(x)}{2^{i}\left[1+p_{i}(x)\right]}}.}
Assume that p ∙ ∙ --> = ( p i ) i = 1 ∞ ∞ --> {\displaystyle p_{\bullet }=\left(p_{i}\right)_{i=1}^{\infty }} is an increasing sequence of seminorms on X {\displaystyle X} and let p {\displaystyle p} be the Fréchet combination of p ∙ ∙ --> . {\displaystyle p_{\bullet }.} Then p {\displaystyle p} is an F-seminorm on X {\displaystyle X} that induces the same locally convex topology as the family p ∙ ∙ --> {\displaystyle p_{\bullet }} of seminorms.[13]
Since p ∙ ∙ --> = ( p i ) i = 1 ∞ ∞ --> {\displaystyle p_{\bullet }=\left(p_{i}\right)_{i=1}^{\infty }} is increasing, a basis of open neighborhoods of the origin consists of all sets of the form { x ∈ ∈ --> X : p i ( x ) < r } {\displaystyle \left\{x\in X~:~p_{i}(x)<r\right\}} as i {\displaystyle i} ranges over all positive integers and r > 0 {\displaystyle r>0} ranges over all positive real numbers.
The translation invariant pseudometric on X {\displaystyle X} induced by this F-seminorm p {\displaystyle p} is d ( x , y ) = ∑ ∑ --> i = 1 ∞ ∞ --> 1 2 i p i ( x − − --> y ) 1 + p i ( x − − --> y ) . {\displaystyle d(x,y)=\sum _{i=1}^{\infty }{\frac {1}{2^{i}}}{\frac {p_{i}(x-y)}{1+p_{i}(x-y)}}.}
This metric was discovered by Fréchet in his 1906 thesis for the spaces of real and complex sequences with pointwise operations.[14]
If each p i {\displaystyle p_{i}} is a paranorm then so is p {\displaystyle p} and moreover, p {\displaystyle p} induces the same topology on X {\displaystyle X} as the family p ∙ ∙ --> {\displaystyle p_{\bullet }} of paranorms.[8] This is also true of the following paranorms on X {\displaystyle X} :
The Fréchet combination can be generalized by use of a bounded remetrization function.
A bounded remetrization function[15] is a continuous non-negative non-decreasing map R : [ 0 , ∞ ∞ --> ) → → --> [ 0 , ∞ ∞ --> ) {\displaystyle R:[0,\infty )\to [0,\infty )} that has a bounded range, is subadditive (meaning that R ( s + t ) ≤ ≤ --> R ( s ) + R ( t ) {\displaystyle R(s+t)\leq R(s)+R(t)} for all s , t ≥ ≥ --> 0 {\displaystyle s,t\geq 0} ), and satisfies R ( s ) = 0 {\displaystyle R(s)=0} if and only if s = 0. {\displaystyle s=0.}
Examples of bounded remetrization functions include arctan --> t , {\displaystyle \arctan t,} tanh --> t , {\displaystyle \tanh t,} t ↦ ↦ --> min { t , 1 } , {\displaystyle t\mapsto \min\{t,1\},} and t ↦ ↦ --> t 1 + t . {\displaystyle t\mapsto {\frac {t}{1+t}}.} [15] If d {\displaystyle d} is a pseudometric (respectively, metric) on X {\displaystyle X} and R {\displaystyle R} is a bounded remetrization function then R ∘ ∘ --> d {\displaystyle R\circ d} is a bounded pseudometric (respectively, bounded metric) on X {\displaystyle X} that is uniformly equivalent to d . {\displaystyle d.} [15]
Suppose that p ∙ ∙ --> = ( p i ) i = 1 ∞ ∞ --> {\displaystyle p_{\bullet }=\left(p_{i}\right)_{i=1}^{\infty }} is a family of non-negative F-seminorm on a vector space X , {\displaystyle X,} R {\displaystyle R} is a bounded remetrization function, and r ∙ ∙ --> = ( r i ) i = 1 ∞ ∞ --> {\displaystyle r_{\bullet }=\left(r_{i}\right)_{i=1}^{\infty }} is a sequence of positive real numbers whose sum is finite. Then p ( x ) := ∑ ∑ --> i = 1 ∞ ∞ --> r i R ( p i ( x ) ) {\displaystyle p(x):=\sum _{i=1}^{\infty }r_{i}R\left(p_{i}(x)\right)} defines a bounded F-seminorm that is uniformly equivalent to the p ∙ ∙ --> . {\displaystyle p_{\bullet }.} [16] It has the property that for any net x ∙ ∙ --> = ( x a ) a ∈ ∈ --> A {\displaystyle x_{\bullet }=\left(x_{a}\right)_{a\in A}} in X , {\displaystyle X,} p ( x ∙ ∙ --> ) → → --> 0 {\displaystyle p\left(x_{\bullet }\right)\to 0} if and only if p i ( x ∙ ∙ --> ) → → --> 0 {\displaystyle p_{i}\left(x_{\bullet }\right)\to 0} for all i . {\displaystyle i.} [16] p {\displaystyle p} is an F-norm if and only if the p ∙ ∙ --> {\displaystyle p_{\bullet }} separate points on X . {\displaystyle X.} [16]
A pseudometric (resp. metric) d {\displaystyle d} is induced by a seminorm (resp. norm) on a vector space X {\displaystyle X} if and only if d {\displaystyle d} is translation invariant and absolutely homogeneous, which means that for all scalars s {\displaystyle s} and all x , y ∈ ∈ --> X , {\displaystyle x,y\in X,} in which case the function defined by p ( x ) := d ( x , 0 ) {\displaystyle p(x):=d(x,0)} is a seminorm (resp. norm) and the pseudometric (resp. metric) induced by p {\displaystyle p} is equal to d . {\displaystyle d.}
If ( X , τ τ --> ) {\displaystyle (X,\tau )} is a topological vector space (TVS) (where note in particular that τ τ --> {\displaystyle \tau } is assumed to be a vector topology) then the following are equivalent:[11]
If ( X , τ τ --> ) {\displaystyle (X,\tau )} is a TVS then the following are equivalent:
Birkhoff–Kakutani theorem — If ( X , τ τ --> ) {\displaystyle (X,\tau )} is a topological vector space then the following three conditions are equivalent:[17][note 1]
By the Birkhoff–Kakutani theorem, it follows that there is an equivalent metric that is translation-invariant.
If ( X , τ τ --> ) {\displaystyle (X,\tau )} is TVS then the following are equivalent:[13]
Let M {\displaystyle M} be a vector subspace of a topological vector space ( X , τ τ --> ) . {\displaystyle (X,\tau ).}
If X {\displaystyle X} is Hausdorff locally convex TVS then X {\displaystyle X} with the strong topology, ( X , b ( X , X ′ ′ --> ) ) , {\displaystyle \left(X,b\left(X,X^{\prime }\right)\right),} is metrizable if and only if there exists a countable set B {\displaystyle {\mathcal {B}}} of bounded subsets of X {\displaystyle X} such that every bounded subset of X {\displaystyle X} is contained in some element of B . {\displaystyle {\mathcal {B}}.} [22]
The strong dual space X b ′ ′ --> {\displaystyle X_{b}^{\prime }} of a metrizable locally convex space (such as a Fréchet space[23]) X {\displaystyle X} is a DF-space.[24] The strong dual of a DF-space is a Fréchet space.[25] The strong dual of a reflexive Fréchet space is a bornological space.[24] The strong bidual (that is, the strong dual space of the strong dual space) of a metrizable locally convex space is a Fréchet space.[26] If X {\displaystyle X} is a metrizable locally convex space then its strong dual X b ′ ′ --> {\displaystyle X_{b}^{\prime }} has one of the following properties, if and only if it has all of these properties: (1) bornological, (2) infrabarreled, (3) barreled.[26]
A topological vector space is seminormable if and only if it has a convex bounded neighborhood of the origin. Moreover, a TVS is normable if and only if it is Hausdorff and seminormable.[14] Every metrizable TVS on a finite-dimensional vector space is a normable locally convex complete TVS, being TVS-isomorphic to Euclidean space. Consequently, any metrizable TVS that is not normable must be infinite dimensional.
If M {\displaystyle M} is a metrizable locally convex TVS that possess a countable fundamental system of bounded sets, then M {\displaystyle M} is normable.[27]
If X {\displaystyle X} is a Hausdorff locally convex space then the following are equivalent:
and if this locally convex space X {\displaystyle X} is also metrizable, then the following may be appended to this list:
In particular, if a metrizable locally convex space X {\displaystyle X} (such as a Fréchet space) is not normable then its strong dual space X b ′ ′ --> {\displaystyle X_{b}^{\prime }} is not a Fréchet–Urysohn space and consequently, this complete Hausdorff locally convex space X b ′ ′ --> {\displaystyle X_{b}^{\prime }} is also neither metrizable nor normable.
Another consequence of this is that if X {\displaystyle X} is a reflexive locally convex TVS whose strong dual X b ′ ′ --> {\displaystyle X_{b}^{\prime }} is metrizable then X b ′ ′ --> {\displaystyle X_{b}^{\prime }} is necessarily a reflexive Fréchet space, X {\displaystyle X} is a DF-space, both X {\displaystyle X} and X b ′ ′ --> {\displaystyle X_{b}^{\prime }} are necessarily complete Hausdorff ultrabornological distinguished webbed spaces, and moreover, X b ′ ′ --> {\displaystyle X_{b}^{\prime }} is normable if and only if X {\displaystyle X} is normable if and only if X {\displaystyle X} is Fréchet–Urysohn if and only if X {\displaystyle X} is metrizable. In particular, such a space X {\displaystyle X} is either a Banach space or else it is not even a Fréchet–Urysohn space.
Suppose that ( X , d ) {\displaystyle (X,d)} is a pseudometric space and B ⊆ ⊆ --> X . {\displaystyle B\subseteq X.} The set B {\displaystyle B} is metrically bounded or d {\displaystyle d} -bounded if there exists a real number R > 0 {\displaystyle R>0} such that d ( x , y ) ≤ ≤ --> R {\displaystyle d(x,y)\leq R} for all x , y ∈ ∈ --> B {\displaystyle x,y\in B} ; the smallest such R {\displaystyle R} is then called the diameter or d {\displaystyle d} -diameter of B . {\displaystyle B.} [14] If B {\displaystyle B} is bounded in a pseudometrizable TVS X {\displaystyle X} then it is metrically bounded; the converse is in general false but it is true for locally convex metrizable TVSs.[14]
Theorem[29] — All infinite-dimensional separable complete metrizable TVS are homeomorphic.
Every topological vector space (and more generally, a topological group) has a canonical uniform structure, induced by its topology, which allows the notions of completeness and uniform continuity to be applied to it. If X {\displaystyle X} is a metrizable TVS and d {\displaystyle d} is a metric that defines X {\displaystyle X} 's topology, then its possible that X {\displaystyle X} is complete as a TVS (i.e. relative to its uniformity) but the metric d {\displaystyle d} is not a complete metric (such metrics exist even for X = R {\displaystyle X=\mathbb {R} } ). Thus, if X {\displaystyle X} is a TVS whose topology is induced by a pseudometric d , {\displaystyle d,} then the notion of completeness of X {\displaystyle X} (as a TVS) and the notion of completeness of the pseudometric space ( X , d ) {\displaystyle (X,d)} are not always equivalent. The next theorem gives a condition for when they are equivalent:
Theorem — If X {\displaystyle X} is a pseudometrizable TVS whose topology is induced by a translation invariant pseudometric d , {\displaystyle d,} then d {\displaystyle d} is a complete pseudometric on X {\displaystyle X} if and only if X {\displaystyle X} is complete as a TVS.[36]
Theorem[37][38] (Klee) — Let d {\displaystyle d} be any[note 2] metric on a vector space X {\displaystyle X} such that the topology τ τ --> {\displaystyle \tau } induced by d {\displaystyle d} on X {\displaystyle X} makes ( X , τ τ --> ) {\displaystyle (X,\tau )} into a topological vector space. If ( X , d ) {\displaystyle (X,d)} is a complete metric space then ( X , τ τ --> ) {\displaystyle (X,\tau )} is a complete-TVS.
Theorem — If X {\displaystyle X} is a TVS whose topology is induced by a paranorm p , {\displaystyle p,} then X {\displaystyle X} is complete if and only if for every sequence ( x i ) i = 1 ∞ ∞ --> {\displaystyle \left(x_{i}\right)_{i=1}^{\infty }} in X , {\displaystyle X,} if ∑ ∑ --> i = 1 ∞ ∞ --> p ( x i ) < ∞ ∞ --> {\displaystyle \sum _{i=1}^{\infty }p\left(x_{i}\right)<\infty } then ∑ ∑ --> i = 1 ∞ ∞ --> x i {\displaystyle \sum _{i=1}^{\infty }x_{i}} converges in X . {\displaystyle X.} [39]
If M {\displaystyle M} is a closed vector subspace of a complete pseudometrizable TVS X , {\displaystyle X,} then the quotient space X / M {\displaystyle X/M} is complete.[40] If M {\displaystyle M} is a complete vector subspace of a metrizable TVS X {\displaystyle X} and if the quotient space X / M {\displaystyle X/M} is complete then so is X . {\displaystyle X.} [40] If X {\displaystyle X} is not complete then M := X , {\displaystyle M:=X,} but not complete, vector subspace of X . {\displaystyle X.}
A Baire separable topological group is metrizable if and only if it is cosmic.[23]
Banach-Saks theorem[45] — If ( x n ) n = 1 ∞ ∞ --> {\displaystyle \left(x_{n}\right)_{n=1}^{\infty }} is a sequence in a locally convex metrizable TVS ( X , τ τ --> ) {\displaystyle (X,\tau )} that converges weakly to some x ∈ ∈ --> X , {\displaystyle x\in X,} then there exists a sequence y ∙ ∙ --> = ( y i ) i = 1 ∞ ∞ --> {\displaystyle y_{\bullet }=\left(y_{i}\right)_{i=1}^{\infty }} in X {\displaystyle X} such that y ∙ ∙ --> → → --> x {\displaystyle y_{\bullet }\to x} in ( X , τ τ --> ) {\displaystyle (X,\tau )} and each y i {\displaystyle y_{i}} is a convex combination of finitely many x n . {\displaystyle x_{n}.}
Mackey's countability condition[14] — Suppose that X {\displaystyle X} is a locally convex metrizable TVS and that ( B i ) i = 1 ∞ ∞ --> {\displaystyle \left(B_{i}\right)_{i=1}^{\infty }} is a countable sequence of bounded subsets of X . {\displaystyle X.} Then there exists a bounded subset B {\displaystyle B} of X {\displaystyle X} and a sequence ( r i ) i = 1 ∞ ∞ --> {\displaystyle \left(r_{i}\right)_{i=1}^{\infty }} of positive real numbers such that B i ⊆ ⊆ --> r i B {\displaystyle B_{i}\subseteq r_{i}B} for all i . {\displaystyle i.}
Generalized series
As described in this article's section on generalized series, for any I {\displaystyle I} -indexed family family ( r i ) i ∈ ∈ --> I {\displaystyle \left(r_{i}\right)_{i\in I}} of vectors from a TVS X , {\displaystyle X,} it is possible to define their sum ∑ ∑ --> i ∈ ∈ --> I r i {\displaystyle \textstyle \sum \limits _{i\in I}r_{i}} as the limit of the net of finite partial sums F ∈ ∈ --> FiniteSubsets --> ( I ) ↦ ↦ --> ∑ ∑ --> i ∈ ∈ --> F r i {\displaystyle F\in \operatorname {FiniteSubsets} (I)\mapsto \textstyle \sum \limits _{i\in F}r_{i}} where the domain FiniteSubsets --> ( I ) {\displaystyle \operatorname {FiniteSubsets} (I)} is directed by ⊆ ⊆ --> . {\displaystyle \,\subseteq .\,} If I = N {\displaystyle I=\mathbb {N} } and X = R , {\displaystyle X=\mathbb {R} ,} for instance, then the generalized series ∑ ∑ --> i ∈ ∈ --> N r i {\displaystyle \textstyle \sum \limits _{i\in \mathbb {N} }r_{i}} converges if and only if ∑ ∑ --> i = 1 ∞ ∞ --> r i {\displaystyle \textstyle \sum \limits _{i=1}^{\infty }r_{i}} converges unconditionally in the usual sense (which for real numbers, is equivalent to absolute convergence). If a generalized series ∑ ∑ --> i ∈ ∈ --> I r i {\displaystyle \textstyle \sum \limits _{i\in I}r_{i}} converges in a metrizable TVS, then the set { i ∈ ∈ --> I : r i ≠ ≠ --> 0 } {\displaystyle \left\{i\in I:r_{i}\neq 0\right\}} is necessarily countable (that is, either finite or countably infinite);[proof 1] in other words, all but at most countably many r i {\displaystyle r_{i}} will be zero and so this generalized series ∑ ∑ --> i ∈ ∈ --> I r i = ∑ ∑ --> r i ≠ ≠ --> 0 i ∈ ∈ --> I r i {\displaystyle \textstyle \sum \limits _{i\in I}r_{i}~=~\textstyle \sum \limits _{\stackrel {i\in I}{r_{i}\neq 0}}r_{i}} is actually a sum of at most countably many non-zero terms.
If X {\displaystyle X} is a pseudometrizable TVS and A {\displaystyle A} maps bounded subsets of X {\displaystyle X} to bounded subsets of Y , {\displaystyle Y,} then A {\displaystyle A} is continuous.[14] Discontinuous linear functionals exist on any infinite-dimensional pseudometrizable TVS.[46] Thus, a pseudometrizable TVS is finite-dimensional if and only if its continuous dual space is equal to its algebraic dual space.[46]
If F : X → → --> Y {\displaystyle F:X\to Y} is a linear map between TVSs and X {\displaystyle X} is metrizable then the following are equivalent:
Open and almost open maps
A vector subspace M {\displaystyle M} of a TVS X {\displaystyle X} has the extension property if any continuous linear functional on M {\displaystyle M} can be extended to a continuous linear functional on X . {\displaystyle X.} [22] Say that a TVS X {\displaystyle X} has the Hahn-Banach extension property (HBEP) if every vector subspace of X {\displaystyle X} has the extension property.[22]
The Hahn-Banach theorem guarantees that every Hausdorff locally convex space has the HBEP. For complete metrizable TVSs there is a converse:
Theorem (Kalton) — Every complete metrizable TVS with the Hahn-Banach extension property is locally convex.[22]
If a vector space X {\displaystyle X} has uncountable dimension and if we endow it with the finest vector topology then this is a TVS with the HBEP that is neither locally convex or metrizable.[22]
Proofs
Lokasi Pengunjung: 3.15.226.33