Metrizable topological vector space

Short description: Topological vector space whose topology can be defined by a metric

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.

## Contents

* 1 Pseudometrics and metrics
* 1.1 Topology induced by a pseudometric
* 2 Pseudometrics and values on topological groups
* 2.1 Translation invariant pseudometrics
* 2.2 Value/G-seminorm
* 2.2.1 Properties of values
* 2.3 Equivalence on topological groups
* 2.4 Pseudometrizable topological groups
* 2.5 An invariant pseudometric that doesn't induce a vector topology
* 3 Additive sequences
* 4 Paranorms
* 4.1 Properties of paranorms
* 4.2 Examples of paranorms
* 5 F-seminorms
* 5.1 F-seminormed spaces
* 5.2 Examples of F-seminorms
* 5.3 Properties of F-seminorms
* 5.4 Topology induced by a single F-seminorm
* 5.5 Topology induced by a family of F-seminorms
* 6 Fréchet combination
* 6.1 As an F-seminorm
* 6.2 As a paranorm
* 6.3 Generalization
* 7 Characterizations
* 7.1 Of (pseudo)metrics induced by (semi)norms
* 7.2 Of pseudometrizable TVS
* 7.3 Of metrizable TVS
* 7.4 Of locally convex pseudometrizable TVS
* 8 Quotients
* 9 Examples and sufficient conditions
* 9.1 Normability
* 10 Metrically bounded sets and bounded sets
* 11 Properties of pseudometrizable TVS
* 11.1 Completeness
* 11.2 Subsets and subsequences
* 11.3 Linear maps
* 11.4 Hahn-Banach extension property
* 12 See also
* 13 Notes
* 14 References
* 15 Bibliography

## Pseudometrics and metrics

A pseudometric on a set X is a map d:X×X->ℝ satisfying the following properties:

1. d(x,x)=0 for all x∈X;
2. Symmetry: d(x,y)=d(y,x) for all x,y∈X;
3. Subadditivity: d(x,z)≤d(x,y)+d(y,z) for all x,y,z∈X.

A pseudometric is called a metric if it satisfies:

4. Identity of indiscernibles: for all x,y∈X, if d(x,y)=0 then x=y.

Ultrapseudometric

A pseudometric d on X is called a ultrapseudometric or a strong pseudometric if it satisfies:

5. Strong/Ultrametric triangle inequality: d(x,z)≤max⁡{d(x,y),d(y,z)} for all x,y,z∈X.

Pseudometric space

A pseudometric space is a pair (X,d) consisting of a set X and a pseudometric d on X such that X's topology is identical to the topology on X induced by d. We call a pseudometric space (X,d) a metric space (resp. ultrapseudometric space) when d is a metric (resp. ultrapseudometric).

### Topology induced by a pseudometric

If d is a pseudometric on a set X then collection of open balls: Br(z):={x∈X:d(x,z)<r} as z ranges over X and r>0 ranges over the positive real numbers, forms a basis for a topology on X that is called the d-topology or the pseudometric topology on X induced by d.

Convention: If (X,d) is a pseudometric space and X is treated as a topological space, then unless indicated otherwise, it should be assumed that X is endowed with the topology induced by d.

Pseudometrizable space

A topological space (X,τ) is called pseudometrizable (resp. metrizable, ultrapseudometrizable) if there exists a pseudometric (resp. metric, ultrapseudometric) d on X such that τ is equal to the topology induced by d.[1]

## Pseudometrics and values on topological groups

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 on a real or complex vector space 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 into a topological vector space).

Every topological vector space (TVS) X is an additive commutative topological group but not all group topologies on X are vector topologies. This is because despite it making addition and negation continuous, a group topology on a vector space 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.

### Translation invariant pseudometrics

If X is an additive group then we say that a pseudometric d on X is translation invariant or just invariant if it satisfies any of the following equivalent conditions:

1. Translation invariance: d(x+z,y+z)=d(x,y) for all x,y,z∈X;
2. d(x,y)=d(x−y,0) for all x,y∈X.

### Value/G-seminorm

If X is a topological group the a value or G-seminorm on X (the G stands for Group) is a real-valued map p:X->ℝ with the following properties:[2]

1. Non-negative: p≥0.
2. Subadditive: p(x+y)≤p(x)+p(y) for all x,y∈X;
3. p(0)=0..
4. Symmetric: p(−x)=p(x) for all x∈X.

where we call a G-seminorm a G-norm if it satisfies the additional condition:

5. Total/Positive definite: If p(x)=0 then x=0.

#### Properties of values

If p is a value on a vector space X then:

* |p(x)−p(y)|≤p(x−y) for all x,y∈X.[3]
* p(nx)≤np(x) and 1np(x)≤p(x/n) for all x∈X and positive integers n.[4]
* The set {x∈X:p(x)=0} is an additive subgroup of X.[3]

### Equivalence on topological groups

Theorem[2] -- Suppose that X is an additive commutative group. If d is a translation invariant pseudometric on X then the map p(x):=d(x,0) is a value on X called the value associated with d, and moreover, d generates a group topology on X (i.e. the d-topology on X makes X into a topological group). Conversely, if p is a value on X then the map d(x,y):=p(x−y) is a translation-invariant pseudometric on X and the value associated with d is just p.

### Pseudometrizable topological groups

Theorem[2] -- If (X,τ) is an additive commutative topological group then the following are equivalent:

1. τ is induced by a pseudometric; (i.e. (X,τ) is pseudometrizable);
2. τ is induced by a translation-invariant pseudometric;
3. the identity element in (X,τ) has a countable neighborhood basis.

If (X,τ) 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.

### An invariant pseudometric that doesn't induce a vector topology

Let X be a non-trivial (i.e. X≠{0}) real or complex vector space and let d be the translation-invariant trivial metric on X defined by d(x,x)=0 and d(x,y)=1 for all x,y∈X such that x≠y. The topology τ that d induces on X is the discrete topology, which makes (X,τ) into a commutative topological group under addition but does not form a vector topology on X because (X,τ) is disconnected but every vector topology is connected. What fails is that scalar multiplication isn't continuous on (X,τ).

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.

## Additive sequences

A collection 𝒩 of subsets of a vector space is called additive[5] if for every N∈𝒩, there exists some U∈𝒩 such that U+U⊆N.

Continuity of addition at 0 -- If (X,+) is a group (as all vector spaces are), τ is a topology on X, and X×X is endowed with the product topology, then the addition map X×X->X (i.e. the map (x,y)↦x+y) is continuous at the origin of X×X if and only if the set of neighborhoods of the origin in (X,τ) 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∙=(Ui)i=0∞ be a collection of subsets of a vector space such that 0∈Ui and Ui+1+Ui+1⊆Ui for all i≥0. For all u∈U0, let 𝕊(u):={n∙=(n1,…,nk):k≥1,ni≥0 for all i, and u∈Un1+⋯+Unk}.

Define f:X->[0,1] by f(x)=1 if x∉U0 and otherwise let f(x):=inf{2−n1+⋯2−nk:n∙=(n1,…,nk)∈𝕊(x)}.

Then f is subadditive (meaning f(x+y)≤f(x)+f(y) for all x,y∈X) and f=0 on ⋂i≥0Ui, so in particular f(0)=0. If all Ui are symmetric sets then f(−x)=f(x) and if all Ui are balanced then f(sx)≤f(x) for all scalars s such that |s|≤1 and all x∈X. If X is a topological vector space and if all Ui are neighborhoods of the origin then f is continuous, where if in addition X is Hausdorff and U∙ forms a basis of balanced neighborhoods of the origin in X then d(x,y):=f(x−y) is a metric defining the vector topology on X.

Proof

Assume that n∙=(n1,…,nk) always denotes a finite sequence of non-negative integers and use the notation: ∑2−n∙:=2−n1+⋯+2−nk and ∑Un∙:=Un1+⋯+Unk.

For any integers n≥0 and d>2, Un⊇Un+1+Un+1⊇Un+1+Un+2+Un+2⊇Un+1+Un+2+⋯+Un+d+Un+d+1+Un+d+1.

From this it follows that if n∙=(n1,…,nk) consists of distinct positive integers then ∑Un∙⊆U−1+min⁡(n∙).

It will now be shown by induction on k that if n∙=(n1,…,nk) consists of non-negative integers such that ∑2−n∙≤2−M for some integer M≥0 then ∑Un∙⊆UM. This is clearly true for k=1 and k=2 so assume that k>2, which implies that all ni are positive. If all ni are distinct then this step is done, and otherwise pick distinct indices i<j such that ni=nj and construct m∙=(m1,…,mk−1) from n∙ by replacing each ni with ni−1 and deleting the jth element of n∙ (all other elements of n∙ are transferred to m∙ unchanged). Observe that ∑2−n∙=∑2−m∙ and ∑Un∙⊆∑Um∙ (because Uni+Unj⊆Uni−1) so by appealing to the inductive hypothesis we conclude that ∑Un∙⊆∑Um∙⊆UM, as desired.

It is clear that f(0)=0 and that 0≤f≤1 so to prove that f is subadditive, it suffices to prove that f(x+y)≤f(x)+f(y) when x,y∈X are such that f(x)+f(y)<1, which implies that x,y∈U0. This is an exercise. If all Ui are symmetric then x∈∑Un∙ if and only if −x∈∑Un∙ from which it follows that f(−x)≤f(x) and f(−x)≥f(x). If all Ui are balanced then the inequality f(sx)≤f(x) for all unit scalars s such that |s|≤1 is proved similarly. Because f is a nonnegative subadditive function satisfying f(0)=0, as described in the article on sublinear functionals, f is uniformly continuous on X if and only if f is continuous at the origin. If all Ui are neighborhoods of the origin then for any real r>0, pick an integer M>1 such that 2−M<r so that x∈UM implies f(x)≤2−M<r. If the set of all Ui form basis of balanced neighborhoods of the origin then it may be shown that for any n>1, there exists some 0<r≤2−n such that f(x)<r implies x∈Un. ◼

## Paranorms

If X is a vector space over the real or complex numbers then a paranorm on X is a G-seminorm (defined above) p:X->ℝ on X that satisfies any of the following additional conditions, each of which begins with "for all sequences x∙=(xi)i=1∞ in X and all convergent sequences of scalars s∙=(si)i=1∞":[6]

1. Continuity of multiplication: if s is a scalar and x∈X are such that p(xi−x)->0 and s∙->s, then p(sixi−sx)->0.
2. Both of the conditions:
* if s∙->0 and if x∈X is such that p(xi−x)->0 then p(sixi)->0;
* if p(x∙)->0 then p(sxi)->0 for every scalar s.
3. Both of the conditions:
* if p(x∙)->0 and s∙->s for some scalar s then p(sixi)->0;
* if s∙->0 then p(six)->0 for all x∈X.
4. Separate continuity:[7]
* if s∙->s for some scalar s then p(sxi−sx)->0 for every x∈X;
* if s is a scalar, x∈X, and p(xi−x)->0 then p(sxi−sx)->0 .

A paranorm is called total if in addition it satisfies:

* Total/Positive definite: p(x)=0 implies x=0.

### Properties of paranorms

If p is a paranorm on a vector space X then the map d:X×X->ℝ defined by d(x,y):=p(x−y) is a translation-invariant pseudometric on X that defines a vector topology on X.[8]

If p is a paranorm on a vector space X then:

* the set {x∈X:p(x)=0} is a vector subspace of X.[8]
* p(x+n)=p(x) for all x,n∈X with p(n)=0.[8]
* If a paranorm p satisfies p(sx)≤|s|p(x) for all x∈X and scalars s, then p is absolutely homogeneity (i.e. equality holds)[8] and thus p is a seminorm.

### Examples of paranorms

* If d is a translation-invariant pseudometric on a vector space X that induces a vector topology τ on X (i.e. (X,τ) is a TVS) then the map p(x):=d(x−y,0) defines a continuous paranorm on (X,τ); moreover, the topology that this paranorm p defines in X is τ.[8]
* If p is a paranorm on X then so is the map q(x):=p(x)/[1+p(x)].[8]
* Every positive scalar multiple of a paranorm (resp. total paranorm) is again such a paranorm (resp. total paranorm).
* Every seminorm is a paranorm.[8]
* The restriction of an paranorm (resp. total paranorm) to a vector subspace is an paranorm (resp. total paranorm).[9]
* The sum of two paranorms is a paranorm.[8]
* If p and q are paranorms on X then so is (p∧q)(x):=inf{p(y)+q(z):x=y+z with y,z∈X}. Moreover, (p∧q)≤p and (p∧q)≤q. This makes the set of paranorms on X into a conditionally complete lattice.[8]
* Each of the following real-valued maps are paranorms on X:=ℝ2:
* (x,y)↦|x|
* (x,y)↦|x|+|y|
* The real-valued maps (x,y)↦|x2−y2| and (x,y)↦|x2−y2|3/2 are not paranorms on X:=ℝ2.[8]
* If x∙=(xi)i∈I is a Hamel basis on a vector space X then the real-valued map that sends x=∑i∈Isixi∈X (where all but finitely many of the scalars si are 0) to ∑i∈I|si| is a paranorm on X, which satisfies p(sx)=|s|p(x) for all x∈X and scalars s.[8]
* The function p(x):=|sin⁡(πx)|+min⁡{2,|x|} is a paranorm on ℝ that is not balanced but nevertheless equivalent to the usual norm on R. Note that the function x↦|sin⁡(πx)| is subadditive.[10]
* Let Xℂ be a complex vector space and let Xℝ denote Xℂ considered as a vector space over ℝ. Any paranorm on Xℂ is also a paranorm on Xℝ.[9]

## F-seminorms

If X is a vector space over the real or complex numbers then an F-seminorm on X (the F stands for Fréchet) is a real-valued map p:X->ℝ with the following four properties: [11]

1. Non-negative: p≥0.
2. Subadditive: p(x+y)≤p(x)+p(y) for all x,y∈X
3. Balanced: p(ax)≤p(x) for x∈X all scalars a satisfying |a|≤1;
* This condition guarantees that each set of the form {z∈X:p(z)≤r} or {z∈X:p(z)<r} for some r≥0 is a balanced set.
4. For every x∈X, p(1nx)->0 as n->∞
* The sequence (1n)n=1∞ can be replaced by any positive sequence converging to the zero.[12]

An F-seminorm is called an F-norm if in addition it satisfies:

5. Total/Positive definite: p(x)=0 implies x=0.

An F-seminorm is called monotone if it satisfies:

6. Monotone: p(rx)<p(sx) for all non-zero x∈X and all real s and t such that s<t.[12]

### F-seminormed spaces

An F-seminormed space (resp. F-normed space)[12] is a pair (X,p) consisting of a vector space X and an F-seminorm (resp. F-norm) p on X.

If (X,p) and (Z,q) are F-seminormed spaces then a map f:X->Z is called an isometric embedding[12] if q(f(x)−f(y))=p(x,y) for all x,y∈X.

Every isometric embedding of one F-seminormed space into another is a topological embedding, but the converse is not true in general.[12]

### Examples of F-seminorms

* Every positive scalar multiple of an F-seminorm (resp. F-norm, seminorm) is again an F-seminorm (resp. F-norm, seminorm).
* The sum of finitely many F-seminorms (resp. F-norms) is an F-seminorm (resp. F-norm).
* If p and q are F-seminorms on X then so is their pointwise supremum x↦sup⁡{p(x),q(x)}. The same is true of the supremum of any non-empty finite family of F-seminorms on X.[12]
* The restriction of an F-seminorm (resp. F-norm) to a vector subspace is an F-seminorm (resp. F-norm).[9]
* A non-negative real-valued function on X is a seminorm if and only if it is a convex F-seminorm, or equivalently, if and only if it is a convex balanced G-seminorm.[10] In particular, every seminorm is an F-seminorm.
* For any 0<p<1, the map f on ℝn defined by [f(x1,…,xn)]p=|x1|p+⋯|xn|p is an F-norm that is not a norm.
* If L:X->Y is a linear map and if q is an F-seminorm on Y, then q∘L is an F-seminorm on X.[12]
* Let Xℂ be a complex vector space and let Xℝ denote Xℂ considered as a vector space over ℝ. Any F-seminorm on Xℂ is also an F-seminorm on Xℝ.[9]

### Properties of F-seminorms

Every F-seminorm is a paranorm and every paranorm is equivalent to some F-seminorm.[7] Every F-seminorm on a vector space X is a value on X. In particular, p(x)=0, and p(x)=p(−x) for all x∈X.

### Topology induced by a single F-seminorm

Theorem[11] -- Let p be an F-seminorm on a vector space X. Then the map d:X×X->ℝ defined by d(x,y):=p(x−y) is a translation invariant pseudometric on X that defines a vector topology τ on X. If p is an F-norm then d is a metric. When X is endowed with this topology then p is a continuous map on X.

The balanced sets {x∈X:p(x)≤r}, as 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}, as r ranges over the positive reals, form a neighborhood basis at the origin for this topology consisting of open sets.

### Topology induced by a family of F-seminorms

Suppose that ℒ is a non-empty collection of F-seminorms on a vector space X and for any finite subset ℱ⊆ℒ and any r>0, let Uℱ,r:=⋂p∈ℱ{x∈X:p(x)<r}.

The set {Uℱ,r:r>0,ℱ⊆ℒ,ℱ finite } forms a filter base on X that also forms a neighborhood basis at the origin for a vector topology on X denoted by τℒ.[12] Each Uℱ,r is a balanced and absorbing subset of X.[12] These sets satisfy[12] Uℱ,r/2+Uℱ,r/2⊆Uℱ,r.

* τℒ is the coarsest vector topology on X making each p∈ℒ continuous.[12]
* τℒ is Hausdorff if and only if for every non-zero x∈X, there exists some p∈ℒ such that p(x)>0.[12]
* If ℱ is the set of all continuous F-seminorms on (X,τℒ) then τℒ=τℱ.[12]
* If ℱ is the set of all pointwise suprema of non-empty finite subsets of ℱ of ℒ then ℱ is a directed family of F-seminorms and τℒ=τℱ.[12]

## Fréchet combination

Suppose that p∙=(pi)i=1∞ is a family of non-negative subadditive functions on a vector space X.

The Fréchet combination[8] of p∙ is defined to be the real-valued map p(x):=∑i=1∞pi(x)2i[1+pi(x)].

### As an F-seminorm

Assume that p∙=(pi)i=1∞ is an increasing sequence of seminorms on X and let p be the Fréchet combination of p∙. Then p is an F-seminorm on X that induces the same locally convex topology as the family p∙ of seminorms.[13]

Since p∙=(pi)i=1∞ is increasing, a basis of open neighborhoods of the origin consists of all sets of the form {x∈X:pi(x)<r} as i ranges over all positive integers and r>0 ranges over all positive real numbers.

The translation invariant pseudometric on X induced by this F-seminorm p is d(x,y)=∑i=1∞12ipi(x−y)1+pi(x−y).

This metric was discovered by Fréchet in his 1906 thesis for the spaces of real and complex sequences with pointwise operations.[14]

### As a paranorm

If each pi is a paranorm then so is p and moreover, p induces the same topology on X as the family p∙ of paranorms.[8] This is also true of the following paranorms on X:

* q(x):=inf{∑i=1npi(x)+1n:n>0 is an integer }.[8]
* r(x):=∑n=1∞min⁡{12n,pn(x)}.[8]

### Generalization

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,∞) that has a bounded range, is subadditive (meaning that R(s+t)≤R(s)+R(t) for all s,t≥0), and satisfies R(s)=0 if and only if s=0.

Examples of bounded remetrization functions include arctan⁡t, tanh⁡t, t↦min⁡{t,1}, and t↦t1+t.[15] If d is a pseudometric (respectively, metric) on X and R is a bounded remetrization function then R∘d is a bounded pseudometric (respectively, bounded metric) on X that is uniformly equivalent to d.[15]

Suppose that p∙=(pi)i=1∞ is a family of non-negative F-seminorm on a vector space X, R is a bounded remetrization function, and r∙=(ri)i=1∞ is a sequence of positive real numbers whose sum is finite. Then p(x):=∑i=1∞riR(pi(x)) defines a bounded F-seminorm that is uniformly equivalent to the p∙.[16] It has the property that for any net x∙=(xa)a∈A in X, p(x∙)->0 if and only if pi(x∙)->0 for all i.[16] p is an F-norm if and only if the p∙ separate points on X.[16]

## Characterizations

### Of (pseudo)metrics induced by (semi)norms

A pseudometric (resp. metric) d is induced by a seminorm (resp. norm) on a vector space X if and only if d is translation invariant and absolutely homogeneous, which means that for all scalars s and all x,y∈X, in which case the function defined by p(x):=d(x,0) is a seminorm (resp. norm) and the pseudometric (resp. metric) induced by p is equal to d.

### Of pseudometrizable TVS

If (X,τ) is a topological vector space (TVS) (where note in particular that τ is assumed to be a vector topology) then the following are equivalent:[11]

1. X is pseudometrizable (i.e. the vector topology τ is induced by a pseudometric on X).
2. X has a countable neighborhood base at the origin.
3. The topology on X is induced by a translation-invariant pseudometric on X.
4. The topology on X is induced by an F-seminorm.
5. The topology on X is induced by a paranorm.

### Of metrizable TVS

If (X,τ) is a TVS then the following are equivalent:

1. X is metrizable.
2. X is Hausdorff and pseudometrizable.
3. X is Hausdorff and has a countable neighborhood base at the origin.[11][12]
4. The topology on X is induced by a translation-invariant metric on X.[11]
5. The topology on X is induced by an F-norm.[11][12]
6. The topology on X is induced by a monotone F-norm.[12]
7. The topology on X is induced by a total paranorm.

Birkhoff–Kakutani theorem -- If (X,τ) is a topological vector space then the following three conditions are equivalent:[17][note 1]

1. The origin {0} is closed in X, and there is a countable basis of neighborhoods for 0 in X.
2. (X,τ) is metrizable (as a topological space).
3. There is a translation-invariant metric on X that induces on X the topology τ, which is the given topology on X.

By the Birkhoff–Kakutani theorem, it follows that there is an equivalent metric that is translation-invariant.

### Of locally convex pseudometrizable TVS

If (X,τ) is TVS then the following are equivalent:[13]

1. X is locally convex and pseudometrizable.
2. X has a countable neighborhood base at the origin consisting of convex sets.
3. The topology of X is induced by a countable family of (continuous) seminorms.
4. The topology of X is induced by a countable increasing sequence of (continuous) seminorms (pi)i=1∞ (increasing means that for all i, pi≥pi+1.
5. The topology of X is induced by an F-seminorm of the form: p(x)=∑n=1∞2−narctan⁡pn(x) where (pi)i=1∞ are (continuous) seminorms on X.[18]

## Quotients

Let M be a vector subspace of a topological vector space (X,τ).

* If X is a pseudometrizable TVS then so is X/M.[11]
* If X is a complete pseudometrizable TVS and M is a closed vector subspace of X then X/M is complete.[11]
* If X is metrizable TVS and M is a closed vector subspace of X then X/M is metrizable.[11]
* If p is an F-seminorm on X, then the map P:X/M->ℝ defined by P(x+M):=inf{p(x+m):m∈M} is an F-seminorm on X/M that induces the usual quotient topology on X/M.[11] If in addition p is an F-norm on X and if M is a closed vector subspace of X then P is an F-norm on X.[11]

## Examples and sufficient conditions

* Every seminormed space (X,p) is pseudometrizable with a canonical pseudometric given by d(x,y):=p(x−y) for all x,y∈X.[19].
* If (X,d) is pseudometric TVS with a translation invariant pseudometric d, then p(x):=d(x,0) defines a paranorm.[20] However, if d is a translation invariant pseudometric on the vector space X (without the addition condition that (X,d) is pseudometric TVS), then d need not be either an F-seminorm[21] nor a paranorm.
* If a TVS has a bounded neighborhood of the origin then it is pseudometrizable; the converse is in general false.[14]
* If a Hausdorff TVS has a bounded neighborhood of the origin then it is metrizable.[14]
* Suppose X is either a DF-space or an LM-space. If X is a sequential space then it is either metrizable or else a Montel DF-space.

If X is Hausdorff locally convex TVS then X with the strong topology, (X,b(X,X′)), is metrizable if and only if there exists a countable set ℬ of bounded subsets of X such that every bounded subset of X is contained in some element of ℬ.[22]

The strong dual space Xb′ of a metrizable locally convex space (such as a Fréchet space[23]) 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 is a metrizable locally convex space then its strong dual Xb′ has one of the following properties, if and only if it has all of these properties: (1) bornological, (2) infrabarreled, (3) barreled.[26]

### Normability

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 is a metrizable locally convex TVS that possess a countable fundamental system of bounded sets, then M is normable.[27]

If X is a Hausdorff locally convex space then the following are equivalent:

1. X is normable.
2. X has a (von Neumann) bounded neighborhood of the origin.
3. the strong dual space Xb′ of X is normable.[28]

and if this locally convex space X is also metrizable, then the following may be appended to this list:

4. the strong dual space of X is metrizable.[28]
5. the strong dual space of X is a Fréchet–Urysohn locally convex space.[23]

In particular, if a metrizable locally convex space X (such as a Fréchet space) is not normable then its strong dual space Xb′ is not a Fréchet–Urysohn space and consequently, this complete Hausdorff locally convex space Xb′ is also neither metrizable nor normable.

Another consequence of this is that if X is a reflexive locally convex TVS whose strong dual Xb′ is metrizable then Xb′ is necessarily a reflexive Fréchet space, X is a DF-space, both X and Xb′ are necessarily complete Hausdorff ultrabornological distinguished webbed spaces, and moreover, Xb′ is normable if and only if X is normable if and only if X is Fréchet–Urysohn if and only if X is metrizable. In particular, such a space X is either a Banach space or else it is not even a Fréchet–Urysohn space.

## Metrically bounded sets and bounded sets

Suppose that (X,d) is a pseudometric space and B⊆X. The set B is metrically bounded or d-bounded if there exists a real number R>0 such that d(x,y)≤R for all x,y∈B; the smallest such R is then called the diameter or d-diameter of B.[14] If B is bounded in a pseudometrizable TVS X then it is metrically bounded; the converse is in general false but it is true for locally convex metrizable TVSs.[14]

## Properties of pseudometrizable TVS

Theorem[29] -- All infinite-dimensional separable complete metrizable TVS are homeomorphic.

* Every metrizable locally convex TVS is a quasibarrelled space,[30] bornological space, and a Mackey space.
* Every complete pseudometrizable TVS is a barrelled space and a Baire space (and hence non-meager).[31] However, there exist metrizable Baire spaces that are not complete.[31]
* If X is a metrizable locally convex space, then the strong dual of X is bornological if and only if it is barreled, if and only if it is infrabarreled.[26]
* If X is a complete pseudometrizable TVS and M is a closed vector subspace of X, then X/M is complete.[11]
* The strong dual of a locally convex metrizable TVS is a webbed space.[32]
* If (X,τ) and (X,ν) are complete metrizable TVSs (i.e. F-spaces) and if ν is coarser than τ then τ=ν;[33] this is no longer guaranteed to be true if any one of these metrizable TVSs is not complete.[34] Said differently, if (X,τ) and (X,ν) are both F-spaces but with different topologies, then neither one of τ and ν contains the other as a subset. One particular consequence of this is, for example, that if (X,p) is a Banach space and (X,q) is some other normed space whose norm-induced topology is finer than (or alternatively, is coarser than) that of (X,p) (i.e. if p≤Cq or if q≤Cp for some constant C>0), then the only way that (X,q) can be a Banach space (i.e. also be complete) is if these two norms p and q are equivalent; if they are not equivalent, then (X,q) can not be a Banach space. As another consequence, if (X,p) is a Banach space and (X,ν) is a Fréchet space, then the map p:(X,ν)->ℝ is continuous if and only if the Fréchet space (X,ν) is the TVS (X,p) (here, the Banach space (X,p) is being considered as a TVS, which means that its norm is "forgetten" but its topology is remembered).
* A metrizable locally convex space is normable if and only if its strong dual space is a Fréchet–Urysohn locally convex space.[23]
* Any product of complete metrizable TVSs is a Baire space.[31]
* A product of metrizable TVSs is metrizable if and only if it all but at most countably many of these TVSs have dimension 0.[35]
* A product of pseudometrizable TVSs is pseudometrizable if and only if it all but at most countably many of these TVSs have the trivial topology.
* Every complete pseudometrizable TVS is a barrelled space and a Baire space (and thus non-meager).[31]
* The dimension of a complete metrizable TVS is either finite or uncountable.[35]

### Completeness

Main article: Complete topological vector space

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 is a metrizable TVS and d is a metric that defines X's topology, then its possible that X is complete as a TVS (i.e. relative to its uniformity) but the metric d is not a complete metric (such metrics exist even for X=ℝ). Thus, if X is a TVS whose topology is induced by a pseudometric d, then the notion of completeness of X (as a TVS) and the notion of completeness of the pseudometric space (X,d) are not always equivalent. The next theorem gives a condition for when they are equivalent:

Theorem -- If X is a pseudometrizable TVS whose topology is induced by a translation invariant pseudometric d, then d is a complete pseudometric on X if and only if X is complete as a TVS.[36]

Theorem[37][38] (Klee) -- Let d be any[note 2] metric on a vector space X such that the topology τ induced by d on X makes (X,τ) into a topological vector space. If (X,d) is a complete metric space then (X,τ) is a complete-TVS.

Theorem -- If X is a TVS whose topology is induced by a paranorm p, then X is complete if and only if for every sequence (xi)i=1∞ in X, if ∑i=1∞p(xi)<∞ then ∑i=1∞xi converges in X.[39]

If M is a closed vector subspace of a complete pseudometrizable TVS X, then the quotient space X/M is complete.[40] If M is a complete vector subspace of a metrizable TVS X and if the quotient space X/M is complete then so is X.[40] If X is not complete then M:=X, but not complete, vector subspace of X.

A Baire separable topological group is metrizable if and only if it is cosmic.[23]

### Subsets and subsequences

* Let M be a separable locally convex metrizable topological vector space and let C be its completion. If S is a bounded subset of C then there exists a bounded subset R of X such that S⊆clCR.[41]
* Every totally bounded subset of a locally convex metrizable TVS X is contained in the closed convex balanced hull of some sequence in X that converges to 0.
* In a pseudometrizable TVS, every bornivore is a neighborhood of the origin.[42]
* If d is a translation invariant metric on a vector space X, then d(nx,0)≤nd(x,0) for all x∈X and every positive integer n.[43]
* If (xi)i=1∞ is a null sequence (that is, it converges to the origin) in a metrizable TVS then there exists a sequence (ri)i=1∞ of positive real numbers diverging to ∞ such that (rixi)i=1∞->0.[43]
* A subset of a complete metric space is closed if and only if it is complete. If a space X is not complete, then X is a closed subset of X that is not complete.
* If X is a metrizable locally convex TVS then for every bounded subset B of X, there exists a bounded disk D in X such that B⊆XD, and both X and the auxiliary normed space XD induce the same subspace topology on B.[44]

Banach-Saks theorem[45] -- If (xn)n=1∞ is a sequence in a locally convex metrizable TVS (X,τ) that converges weakly to some x∈X, then there exists a sequence y∙=(yi)i=1∞ in X such that y∙->x in (X,τ) and each yi is a convex combination of finitely many xn.

Mackey's countability condition[14] -- Suppose that X is a locally convex metrizable TVS and that (Bi)i=1∞ is a countable sequence of bounded subsets of X. Then there exists a bounded subset B of X and a sequence (ri)i=1∞ of positive real numbers such that Bi⊆riB for all i.

Generalized series

As described in this article's section on generalized series, for any I-indexed family family (ri)i∈I of vectors from a TVS X, it is possible to define their sum ∑i∈Iri as the limit of the net of finite partial sums F∈FiniteSubsets⁡(I)↦∑i∈Fri where the domain FiniteSubsets⁡(I) is directed by ⊆. If I=ℕ and X=ℝ, for instance, then the generalized series ∑i∈ℕri converges if and only if ∑i=1∞ri converges unconditionally in the usual sense (which for real numbers, is equivalent to absolute convergence). If a generalized series ∑i∈Iri converges in a metrizable TVS, then the set {i∈I:ri≠0} is necessarily countable (that is, either finite or countably infinite);[proof 1] in other words, all but at most countably many ri will be zero and so this generalized series ∑i∈Iri=∑ri≠0i∈Iri is actually a sum of at most countably many non-zero terms.

### Linear maps

If X is a pseudometrizable TVS and A maps bounded subsets of X to bounded subsets of Y, then 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 is a linear map between TVSs and X is metrizable then the following are equivalent:

1. F is continuous;
2. F is a (locally) bounded map (that is, F maps (von Neumann) bounded subsets of X to bounded subsets of Y);[12]
3. F is sequentially continuous;[12]
4. the image under F of every null sequence in X is a bounded set[12] where by definition, a null sequence is a sequence that converges to the origin.
5. F maps null sequences to null sequences;

Open and almost open maps

Theorem: If X is a complete pseudometrizable TVS, Y is a Hausdorff TVS, and T:X->Y is a closed and almost open linear surjection, then T is an open map.[47]

Theorem: If T:X->Y is a surjective linear operator from a locally convex space X onto a barrelled space Y (e.g. every complete pseudometrizable space is barrelled) then T is almost open.[47]

Theorem: If T:X->Y is a surjective linear operator from a TVS X onto a Baire space Y then T is almost open.[47]

Theorem: Suppose T:X->Y is a continuous linear operator from a complete pseudometrizable TVS X into a Hausdorff TVS Y. If the image of T is non-meager in Y then T:X->Y is a surjective open map and Y is a complete metrizable space.[47]

### Hahn-Banach extension property

Main article: Hahn-Banach theorem

A vector subspace M of a TVS X has the extension property if any continuous linear functional on M can be extended to a continuous linear functional on X.[22] Say that a TVS X has the Hahn-Banach extension property (HBEP) if every vector subspace of 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 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]

## See also

* Asymmetric norm - Generalization of the concept of a norm
* Complete metric space - Metric geometry
* Complete topological vector space - Structure in functional analysis
* Equivalence of metrics
* F-space - Topological vector space with a complete translation-invariant metric
* Fréchet space - A locally convex topological vector space that is also a complete metric space
* Generalised metric - Metric geometry
* K-space (functional analysis)
* Locally convex topological vector space - Vector space with a topology defined by convex open sets
* Metric space - Mathematical space with a notion of distance
* Pseudometric space - Generalization of metric spaces in mathematics
* Seminorm
* Sublinear function - Type of function in linear algebra
* Uniform space - Topological space with a notion of uniform properties
* Ursescu theorem - Generalization of closed graph, open mapping, and uniform boundedness theorem

## Notes

1. ↑ In fact, this is true for topological group, for the proof doesn't use the scalar multiplications.
2. ↑ Not assumed to be translation-invariant.

Proofs

1. ↑ Suppose the net ∑i∈Iri=deflim⁡A∈FiniteSubsets⁡(I) ∑i∈Ari=lim⁡{∑i∈Ari:A⊆I,A finite } converges to some point in a metrizable TVS X, where recall that this net's domain is the directed set (FiniteSubsets⁡(I),⊆). Like every convergent net, this convergent net of partial sums A↦∑i∈Ari is a Cauchy net, which for this particular net means (by definition) that for every neighborhood W of the origin in X, there exists a finite subset A0 of I such that ∑i∈Bri−∑i∈Cri∈W for all finite supersets B,C⊇A0; this implies that ri∈W for every i∈I∖A0 (by taking B:=A0∪{i} and C:=A0). Since X is metrizable, it has a countable neighborhood basis U1,U2,… at the origin, whose intersection is necessarily U1∩U2∩⋯={0} (since X is a Hausdorff TVS). For every positive integer n∈ℕ, pick a finite subset An⊆I such that ri∈Un for every i∈I∖An. If i belongs to (I∖A1)∩(I∖A2)∩⋯=I∖(A1∪A2∪⋯) then ri belongs to U1∩U2∩⋯={0}. Thus ri=0 for every index i∈I that does not belong to the countable set A1∪A2∪⋯. ◼

## References

1. ↑ Narici & Beckenstein 2011, pp. 1–18.
2. ↑ 2.0 2.1 2.2 Narici & Beckenstein 2011, pp. 37–40.
3. ↑ 3.0 3.1 Swartz 1992, p. 15.
4. ↑ Wilansky 2013, p. 17.
5. ↑ 5.0 5.1 Wilansky 2013, pp. 40–47.
6. ↑ Wilansky 2013, p. 15.
7. ↑ 7.0 7.1 Schechter 1996, pp. 689–691.
8. ↑ 8.00 8.01 8.02 8.03 8.04 8.05 8.06 8.07 8.08 8.09 8.10 8.11 8.12 8.13 8.14 Wilansky 2013, pp. 15–18.
9. ↑ 9.0 9.1 9.2 9.3 Schechter 1996, p. 692.
10. ↑ 10.0 10.1 Schechter 1996, p. 691.
11. ↑ 11.00 11.01 11.02 11.03 11.04 11.05 11.06 11.07 11.08 11.09 11.10 11.11 Narici & Beckenstein 2011, pp. 91–95.
12. ↑ 12.00 12.01 12.02 12.03 12.04 12.05 12.06 12.07 12.08 12.09 12.10 12.11 12.12 12.13 12.14 12.15 12.16 12.17 12.18 12.19 Jarchow 1981, pp. 38–42.
13. ↑ 13.0 13.1 Narici & Beckenstein 2011, p. 123.
14. ↑ 14.0 14.1 14.2 14.3 14.4 14.5 14.6 14.7 Narici & Beckenstein 2011, pp. 156–175.
15. ↑ 15.0 15.1 15.2 Schechter 1996, p. 487.
16. ↑ 16.0 16.1 16.2 Schechter 1996, pp. 692–693.
17. ↑ Köthe 1983, section 15.11
18. ↑ Schechter 1996, p. 706.
19. ↑ Narici & Beckenstein 2011, pp. 115–154.
20. ↑ Wilansky 2013, pp. 15–16.
21. ↑ Schaefer & Wolff 1999, pp. 91–92.
22. ↑ 22.0 22.1 22.2 22.3 22.4 Narici & Beckenstein 2011, pp. 225–273.
23. ↑ 23.0 23.1 23.2 23.3 Gabriyelyan, S.S. "On topological spaces and topological groups with certain local countable networks (2014)
24. ↑ 24.0 24.1 Schaefer & Wolff 1999, p. 154.
25. ↑ Schaefer & Wolff 1999, p. 196.
26. ↑ 26.0 26.1 26.2 Schaefer & Wolff 1999, p. 153.
27. ↑ Schaefer & Wolff 1999, pp. 68–72.
28. ↑ 28.0 28.1 Trèves 2006, p. 201.
29. ↑ Wilansky 2013, p. 57.
30. ↑ Jarchow 1981, p. 222.
31. ↑ 31.0 31.1 31.2 31.3 Narici & Beckenstein 2011, pp. 371–423.
32. ↑ Narici & Beckenstein 2011, pp. 459–483.
33. ↑ Köthe 1969, p. 168.
34. ↑ Wilansky 2013, p. 59.
35. ↑ 35.0 35.1 Schaefer & Wolff 1999, pp. 12–35.
36. ↑ Narici & Beckenstein 2011, pp. 47–50.
37. ↑ Schaefer & Wolff 1999, p. 35.
38. ↑ Klee, V. L. (1952). "Invariant metrics in groups (solution of a problem of Banach)". Proc. Amer. Math. Soc. 3 (3): 484–487. doi:10.1090/s0002-9939-1952-0047250-4. https://www.ams.org/journals/proc/1952-003-03/S0002-9939-1952-0047250-4/S0002-9939-1952-0047250-4.pdf.
39. ↑ Wilansky 2013, pp. 56–57.
40. ↑ 40.0 40.1 Narici & Beckenstein 2011, pp. 47–66.
41. ↑ Schaefer & Wolff 1999, pp. 190–202.
42. ↑ Narici & Beckenstein 2011, pp. 172–173.
43. ↑ 43.0 43.1 Rudin 1991, p. 22.
44. ↑ Narici & Beckenstein 2011, pp. 441–457.
45. ↑ Rudin 1991, p. 67.
46. ↑ 46.0 46.1 Narici & Beckenstein 2011, p. 125.
47. ↑ 47.0 47.1 47.2 47.3 Narici & Beckenstein 2011, pp. 466–468.

## Bibliography

* Jarchow, Hans (1981). Locally convex spaces. Stuttgart: B.G. Teubner. ISBN 978-3-519-02224-4. OCLC 8210342.
* Bourbaki, Nicolas (1987). Topological Vector Spaces: Chapters 1–5. Éléments de mathématique. 2. Berlin New York: Springer-Verlag. ISBN 978-3-540-42338-6. OCLC 17499190. http://www.numdam.org/item?id=AIF_1950__2__5_0.
* Bourbaki, Nicolas (1950). "Sur certains espaces vectoriels topologiques" (in French). Annales de l'Institut Fourier 2: 5–16 (1951). doi:10.5802/aif.16.
* Edwards, Robert E. (Jan 1, 1995). Functional Analysis: Theory and Applications. New York: Dover Publications. ISBN 978-0-486-68143-6. OCLC 30593138.
* Grothendieck, Alexander (January 1, 1973). Topological Vector Spaces. New York: Gordon and Breach Science Publishers. ISBN 978-0-677-30020-7. OCLC 886098. https://archive.org/details/topologicalvecto0000grot.
* Jarchow, Hans (1981). Locally convex spaces. Stuttgart: B.G. Teubner. ISBN 978-3-519-02224-4. OCLC 8210342.
* Khaleelulla, S. M. (July 1, 1982). written at Berlin Heidelberg. Counterexamples in Topological Vector Spaces. Lecture Notes in Mathematics. 936. Berlin New York: Springer-Verlag. ISBN 978-3-540-11565-6. OCLC 8588370.
* Köthe, Gottfried (1969). Topological Vector Spaces I. Grundlehren der mathematischen Wissenschaften. 159. New York: Springer Science & Business Media. ISBN 978-3-642-64988-2. OCLC 840293704.
* Köthe, Gottfried (1979). Topological Vector Spaces II. Grundlehren der mathematischen Wissenschaften. 237. New York: Springer Science & Business Media. ISBN 978-0-387-90400-9. OCLC 180577972.
* Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834.
* Rudin, Walter (January 1, 1991). Functional Analysis. International Series in Pure and Applied Mathematics. 8 (Second ed.). New York, NY: McGraw-Hill Science/Engineering/Math. ISBN 978-0-07-054236-5. OCLC 21163277. https://archive.org/details/functionalanalys00rudi.
* Robertson, Alex P.; Robertson, Wendy J. (1980). Topological Vector Spaces. Cambridge Tracts in Mathematics. 53. Cambridge England: Cambridge University Press. ISBN 978-0-521-29882-7. OCLC 589250.
* Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.
* Schechter, Eric (1996). Handbook of Analysis and Its Foundations. San Diego, CA: Academic Press. ISBN 978-0-12-622760-4. OCLC 175294365.
* Swartz, Charles (1992). An introduction to Functional Analysis. New York: M. Dekker. ISBN 978-0-8247-8643-4. OCLC 24909067.
* Trèves, François (August 6, 2006). Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.
* Wilansky, Albert (2013). Modern Methods in Topological Vector Spaces. Mineola, New York: Dover Publications, Inc. ISBN 978-0-486-49353-4. OCLC 849801114.
* Husain, Taqdir (1978). Barrelledness in topological and ordered vector spaces. Berlin New York: Springer-Verlag. ISBN 3-540-09096-7. OCLC 4493665.

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Functional analysis (topics)

Topological vector spaces|

* Asplund
* Banach (list)
* Banach lattice
* Barrelled
* Bornological
* Brauner
* F-space
* Fréchet (tame)
* Hilbert (Inner product space
* Polarization identity)
* LF-space
* Locally convex (Seminorms/Minkowski functionals)
* Mackey
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* Quasinormed
* Reflexive
* Riesz
* Smith
* Stereotype
* Strictly convex
* Webbed
* Topological tensor product (of Hilbert spaces)

Topologies of function spaces|

* Dual
* Dual space (Dual norm)
* Operator
* Ultraweak
* Weak (polar
* operator)
* Mackey
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* operator)
* Ultrastrong
* Uniform convergence

Linear operators|

* Adjoint
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* operator
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Operator theory|

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Theorems|

* Banach–Alaoglu
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Analysis|

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