Metrizable topological vector space

In functional analysis and related areas of mathematics, a metrizable (resp. pseudometrizable) topological vector spaces (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.

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:

  1. 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:

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

  1. 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 1/np(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.

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 (i.e. f (x + y) ≤ f (x) + f (y) for all x, y ∈ X) and f = 0 on ∩i ≥ 0 Ui, 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 (s x) ≤ 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

We also assume that n• = (n1, ⋅⋅⋅, nk) always denotes a finite sequence of non-negative integers and we will 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•).

We show 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 we're done, otherwise pick distinct indices i < j such that ni = nj and construct m• = (m1, ..., mk-1) from n• by replacing 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• (since 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 (s x) ≤ f (x) for all unit scalars s is proved similarly. Since f is a nonnegative subadditive function satisfying f (0) = 0, f is uniformly continuous on X if and only if f is continuous at 0. 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 all Ui form basis of balanced neighborhoods of the origin then one may show that for any n > 0, 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(si xi - sx) → 0.
  2. Both of the conditions:
    • if s• → 0 and if x ∈ X is such that p(xi - x) → 0 then p(si xi) → 0;
    • if p(xi) → 0 then p(s xi) → 0 for every scalar s.
  3. Both of the conditions:
    • if p(xi) → 0 and s• → s for some scalar s then p(si xi) → 0;
    • if s• → 0 then p(si x) → 0 for all x ∈ X.
  4. Separate continuity:[7]
    • if s• → s for some scalar s then p(si x - sx) → 0 for every x ∈ X;
    • if s is a scalar, x ∈ X, and p(xi - x) → 0 then p(s xi - 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 map (x, y) ↦ √x2 + y2 is 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 ∈ I sixi ∈ 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 vctor 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 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 all x ∈ X and all scalars a satisfying |a| ≤ 1 ;
    • This condition guarantees that each set of the form { x ∈ X : p(x) ≤ r } or { x ∈ X : p(x) < r } for some r ≥ 0 is balanced.
  4. for every x ∈ X, p(1/n x) → 0 as n → ∞
    • The sequence (1/n)∞
      n=1       can be replaced by any positive sequence converging to 0.[12]

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

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

An F-seminorm is called monotone if it satisfies:

  1. 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 vctor 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(0) = 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 𝒮 is a non-empty collection of F-seminorms on a vector space X and for any finite subset ℱ ⊆ 𝒮 and any r > 0, let

Uℱ, r = ∩p ∈ ℱ { x ∈ X : p(x) < r  }.

The set { Uℱ, r : r > 0, ℱ ⊆ 𝒮, 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 τ𝒮.[12] Each Uℱ, r is a balanced and absorbing subset of X.[12] These sets satisfy

Uℱ, r/2 + Uℱ, r/2 ⊆ Uℱ, r.[12]
  • τ𝒮 is the coarsest vector topology on X making each p ∈ 𝒮 continuous.[12]
  • τ𝒮 is Hausdorff if and only if for every non-zero x ∈ X, there exists some p ∈ 𝒮 such that p(x) > 0.[12]
  • If 𝒯 is the set of all continuous F-seminorms on (X, τ𝒮) then τ𝒮 = τ𝒯.[12]
  • If 𝒯 is the set of all pointwise suprema of non-empty finite subsets of ℱ of 𝒮 then 𝒯 is a directed family of F-seminorms and τ𝒮 = τ𝒯.[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

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

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 { ∑n
    i=1     pi(x) + 1/n : n > 0 is an integer     }.[8]
  • r(x) := ∑∞
    n=1     min { 1/2n, 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 is subadditive (i.e. R (s + t) ≤ R (s) + R (t) for all s, t ≥ 0), has a bounded range, 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 ↦ t/1 + t.[15] If d is a pseudometric (resp. metric) on X and R} is a bounded remetrization function then R ∘ d is a bounded pseudometric (resp. 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

defines a bounded F-seminorm that is uniformly equivalent to the p•.[16] It has the property that for any net x• = (xi)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 d(sx, sy) = |s| d(x, y) 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:
    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 0 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]

Normability

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

  1. X is normable.
  2. X has a bounded neighborhood of the origin.
  3. the strong dual of X is normable.[23]
  4. the strong dual of X is metrizable.[23]

Moreover, if M is a locally convex metrizable topological vector space that possess a countable fundamental system of bounded sets, then M is normable.[24] If a TVS X has a convex bounded neighborhood of the origin then it is seminormable; if in addition X is Hausdorff then it is normable.[14]

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[25] — All infinite-dimensional separable complete metrizable TVS are homeomorphic.

  • Every metrizable locally convex TVS is a quasibarrelled space,[26] bornological space, and a Mackey space.
  • Every complete pseudometrizable TVS is a barrelled space and a Baire space (and hence non-meager).[27] However, there exist metrizable Baire spaces that are not complete.[27]
  • If X is a metrizable locally convex space, then the strong dual of X is bornological if and only if it is infrabarreled, if and only if it is barreled.[28]
  • If X is a complete pseudo-metrizable 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.[29]
  • If (X, 𝜏) and (X, 𝜐) are complete metrizable TVSs and if 𝜐 is coarser than 𝜏 then 𝜏 = 𝜐;[30] this is no longer true if either one of these metrizable TVSs is not complete.[31]
  • A metrizable locally convex space is normable if and only if its strong dual space is a Frechet-Urysohn locally convex space.[32]
  • Any product of complete metrizable TVSs is a Baire space.[27]
  • A product of metrizable TVSs is metrizable if and only if it all but at most countably many of these TVSs have dimension 0.[33]
  • A product of pseudometrizable TVSs is pseduometrizable 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).[27]
  • The dimension of a complete metrizable TVS is either finite or uncountable.[33]

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.[34]

Theorem[35][36] (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.[37]

If M is a closed vector subspace of a complete pseudometrizable TVS X, then the quotient space X ∖ M is complete.[38] 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.[38] If X is not complete then M := X is a closed, but not complete, vector subspace of X.

A Baire separable topological group is metrizable if and only if it is cosmic.[32]

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 ⊆ clC R.[39]
  • 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.[40]
  • 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.[41]
  • If (xi)∞
    i=1     is a null sequence (i.e. 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    .[41]
  • 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.[42]

Banach-Saks theorem[43] — 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 ⊆ ri B for all i.

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.[44] Thus, a pseudometrizable TVS is finite-dimensional if and only if its continuous dual space is equal to its algebraic dual space.[44]

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 (i.e. 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.[45]
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.[45]
Theorem: If T : X → Y is a surjective linear operator from a TVS X onto a Baire space Y then T is almost open.[45]
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.[45]

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

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.

References
  1. Narici & Beckenstein 2011, pp. 1-18.
  2. Narici & Beckenstein 2011, pp. 37-40.
  3. Swartz 1992, p. 15.
  4. Wilansky 2013, p. 17.
  5. Wilansky 2013, pp. 40-47.
  6. Wilansky 2013, p. 15.
  7. Schechter 1996, pp. 689-691.
  8. Wilansky 2013, pp. 15-18.
  9. Schechter 1996, p. 692.
  10. Schechter 1996, p. 691.
  11. Narici & Beckenstein 2011, pp. 91-95.
  12. Jarchow 1981, pp. 38-42.
  13. Narici & Beckenstein 2011, p. 123.
  14. Narici & Beckenstein 2011, pp. 156-175.
  15. Schechter 1996, p. 487.
  16. 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. Narici & Beckenstein 2011, pp. 225-273.
  23. Trèves 2006, p. 201.
  24. Schaefer & Wolff 1999, pp. 68-72.
  25. Wilansky 2013, p. 57.
  26. Jarchow 1981, p. 222.
  27. Narici & Beckenstein 2011, pp. 371-423.
  28. Schaefer & Wolff 1999, p. 153.
  29. Narici & Beckenstein 2011, pp. 459-483.
  30. Köthe 1969, p. 168.
  31. Wilansky 2013, p. 59.
  32. Gabriyelyan, S.S. "On topological spaces and topological groups with certain local countable networks (2014)
  33. Schaefer & Wolff 1999, pp. 12-35.
  34. Narici & Beckenstein 2011, pp. 47-50.
  35. Schaefer & Wolff 1999, p. 35.
  36. Klee, V. L. (1952). "Invariant metrics in groups (solution of a problem of Banach)" (PDF). Proc. Amer. Math. Soc. 3 (3): 484–487. doi:10.1090/s0002-9939-1952-0047250-4.
  37. Wilansky 2013, pp. 56-57.
  38. Narici & Beckenstein 2011, pp. 47-66.
  39. Schaefer & Wolff 1999, pp. 190-202.
  40. Narici & Beckenstein 2011, pp. 172-173.
  41. Rudin 1991, p. 22.
  42. Narici & Beckenstein 2011, pp. 441-457.
  43. Rudin 1991, p. 67.
  44. Narici & Beckenstein 2011, p. 125.
  45. Narici & Beckenstein 2011, pp. 466-468.

Bibliography
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