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 LMspace is an inductive limit of a sequence of locally convex metrizable TVS.
Pseudometrics and metricsEdit
A pseudometric on a set X is a map d : X × X → ℝ satisfying the following properties:
 d(x, x) = 0 for all x ∈ X;
 Symmetry: d(x, y) = d(y, x) for all x, y ∈ X;
 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:
 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:
 Strong/Ultrametric triangle inequality: for all x, y ∈ X, d(x, y) ≤ max { d(x, z), d(y, z)}.
 Pseudometric space
 Definition: 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 pseudometricEdit
If d is a pseudometric on a set X then collection of open balls:
 B_{r}(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 dtopology 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
 Definition: 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 groupsEdit
 Definition: An additive topological group is an additive group endowed with a topology, called a group topology, under which addition and negation become continuous operators.
 Definition: 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 nontrivial vector space makes addition and negation continuous but do not make scalar multiplication continuous.
Translation invariant pseudometricsEdit
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:
 Translation invariance: d(x + z, y + z) = d(x, y) for all x, y, z ∈ X;
 d(x, y) = d(x  y, 0) for all x, y ∈ X.
Value/GseminormEdit
If X is a topological group the a value or Gseminorm on X (the G stands for Group) is a realvalued map p : X → ℝ with the following properties:^{[2]}
 Nonnegative: p ≥ 0.
 Subadditive: p(x+y) ≤ p(x) + p(y) for all x, y ∈ X;
 p(0) = 0.
 Symmetric: p(x) = p(x) for all x ∈ X.
where we call a Gseminorm a Gnorm if it satisfies the additional condition:
 Total/Positive definite: If p(x) = 0 then x = 0.
Properties of valuesEdit
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 groupsEdit
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 dtopology 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 translationinvariant pseudometric on X and the value associated with d is just p.
Pseudometrizable topological groupsEdit
Theorem^{[2]} — If (X, τ) is an additive commutative topological group then the following are equivalent:
 τ is induced by a pseudometric; (i.e. (X, τ) is pseudometrizable);
 τ is induced by a translationinvariant pseudometric;
 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 topologyEdit
Let X be a nontrivial (i.e. X ≠ { 0 }) real or complex vector space and let d be the translationinvariant 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 translationinvariant (pseudo)metric is not enough to guarantee a vector topology, which leads us to define paranorms and Fseminorms.
Additive sequencesEdit
Definition:^{[5]} A collection 𝒩 of subsets of a vector space is called additive if for every N ∈ 𝒩, there exists some U ∈ 𝒩 such that U + U ⊆ N.
Continuity of addition at 0^{[5]} — 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."
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 nonnegative continuous realvalued 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_{•} = (U_{i})^{∞}
_{i=0} be a collection of subsets of a vector space such that 0 ∈ U_{i} and U_{i+1} + U_{i+1} ⊆ U_{i} for all i ≥ 0.
For all u ∈ U_{0}, let
 𝕊(u) := { n_{•} = (n_{1}, ⋅⋅⋅, n_{k}) : k ≥ 1, n_{i} ≥ 0 for all i, and u ∈ U_{n1} + ⋅⋅⋅ + U_{nk} }.
Define f : X → [0, 1] by f (x) = 1 if x ∉ U_{0} and otherwise let
 f (x) := inf { 2^{ n1} + ⋅⋅⋅ + 2^{ nk} : n_{•} = (n_{1}, ⋅⋅⋅, n_{k}) ∈ 𝕊(x) }.
Then f is subadditive (i.e. f (x + y) ≤ f (x) + f (y) for all x, y ∈ X) and f = 0 on U_{i}, so in particular f (0) = 0. If all U_{i} are symmetric sets then f ( x) = f (x) and if all U_{i} 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 U_{i} 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_{•} = (n_{1}, ⋅⋅⋅, n_{k}) always denotes a finite sequence of nonnegative integers and we will use the notation:
Observe that for any integers n ≥ 0 and d > 2,
From this it follows that if n_{•} = (n_{1}, ⋅⋅⋅, n_{k}) consists of distinct positive integers then ∑ U_{n•} ⊆ U_{1 + min (n•)}. We show by induction on k that if n_{•} = (n_{1}, ⋅⋅⋅, n_{k}) consists of nonnegative integers such that ∑ 2^{ n•} ≤ 2^{ M} for some integer M ≥ 0 then ∑ U_{n•} ⊆ U_{M}. This is clearly true for k = 1 and k = 2 so assume that k > 2, which implies that all n_{i} are positive. If all n_{i} are distinct then we're done, otherwise pick distinct indices i < j such that n_{i} = n_{j} and construct m_{•} = (m_{1}, ..., m_{k1}) from n_{•} by replacing n_{i} with n_{i}  1 and deleting the j^{th} element of n_{•} (all other elements of n_{•} are transferred to m_{•} unchanged). Observe that ∑ 2^{ n•} = ∑ 2^{ m•} and ∑ U_{n•} ⊆ ∑ U_{m•} (since U_{ni} + U_{nj} ⊆ U_{ni  1}) so by appealing to the inductive hypothesis we conclude that ∑ U_{n•} ⊆ ∑ U_{m•} ⊆ U_{M}, 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 ∈ U_{0}. This is an exercise. If all U_{i} are symmetric then x ∈ ∑ U_{n•} if and only if  x ∈ ∑ U_{n•} from which it follows that f (x) ≤ f (x) and f (x) ≥ f (x). If all U_{i} 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 U_{i} are neighborhoods of the origin then for any real r > 0, pick an integer M > 1 such that 2^{ M} < r so that x ∈ U_{M} implies f (x) ≤ 2^{ M} < r. If all U_{i} 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 ∈ U_{n}. ∎ 
ParanormsEdit
If X is a vector space over the real or complex numbers then a paranorm on X is a Gseminorm (defined above) p : X → ℝ on X that satisfies any of the following additional conditions, each of which begins with "for all sequences x_{•} = (x_{i})^{∞}
_{i=1} in X and all convergent sequences of scalars s_{•} = (s_{i})^{∞}
_{i=1}":^{[6]}
 Continuity of multiplication: if s is a scalar and x ∈ X are such that p(x_{i}  x) → 0 and s_{•} → s, then p(s_{i} x_{i}  sx) → 0.
 Both of the conditions:
 if s_{•} → 0 and if x ∈ X is such that p(x_{i}  x) → 0 then p(s_{i} x_{i}) → 0;
 if p(x_{i}) → 0 then p(s x_{i}) → 0 for every scalar s.
 Both of the conditions:
 if p(x_{i}) → 0 and s_{•} → s for some scalar s then p(s_{i} x_{i}) → 0;
 if s_{•} → 0 then p(s_{i} x) → 0 for all x ∈ X.
 Separate continuity:^{[7]}
 if s_{•} → s for some scalar s then p(s_{i} x  sx) → 0 for every x ∈ X;
 if s is a scalar, x ∈ X, and p(x_{i}  x) → 0 then p(s x_{i}  sx) → 0.
A paranorm is called total if in addition it satisfies:
 Total/Positive definite: p(x) = 0 implies x = 0.
Properties of paranormsEdit
 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 translationinvariant 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 paranormsEdit
 If d is a translationinvariant 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 realvalued maps are paranorms on X := ℝ^{2}:
 (x, y) ↦ x
 (x, y) ↦ x + y
 The realvalued map (x, y) ↦ √x^{2} + y^{2} is not paranorms on X := ℝ^{2}.^{[8]}
 If x_{•} = (x_{i})_{i ∈ I} is a Hamel basis on a vector space X then the realvalued map that sends x = ∑_{i ∈ I} s_{i}x_{i} ∈ X (where all but finitely many of the scalars s_{i} are 0) to ∑_{i ∈ I} √s_{i} 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 X_{ℂ} be a complex vector space and let X_{ℝ} denote X_{ℂ} considered as a vctor space over ℝ. Any paranorm on X_{ℂ} is also a paranorm on X_{ℝ}.^{[9]}
FseminormsEdit
If X is a vector space over the real or complex numbers then an Fseminorm on X (the F stands for Fréchet) is a realvalued map p : X → ℝ with the following properties:^{[11]}
 Nonnegative: p ≥ 0.
 Subadditive: p(x+y) ≤ p(x) + p(y) for all x, y ∈ X;
 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.
 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]}
 The sequence (1/n)^{∞}
An Fseminorm is called an Fnorm if in addition it satisfies:
 Total/Positive definite: p(x) = 0 implies x = 0.
An Fseminorm is called monotone if it satisfies:
 Monotone: p(rx) < p(sx) for all nonzero x ∈ X and all real s and t such that s < t.^{[12]}
Fseminormed spacesEdit
 Definition:^{[12]} An Fseminormed space (resp. Fnormed space) is a pair (X, p) consisting of a vector space X and an Fseminorm (resp. Fnorm) p on X.
 Definition:^{[12]} If (X, p) and (Z, q) are Fseminormed spaces then a map f : X → Z is called an isometric embedding if q(f (x)  f (y)) = p(x  y) for all x, y ∈ X.
Every isometric embedding of one Fseminormed space into another is a topological embedding, but the converse is not true in general.^{[12]}
Examples of FseminormsEdit
 Every positive scalar multiple of an Fseminorm (resp. Fnorm, seminorm) is again an Fseminorm (resp. Fnorm, seminorm).
 The sum of finitely many Fseminorms (resp. Fnorms) is an Fseminorm (resp. Fnorm).
 If p and q are Fseminorms on X then so is their pointwise supremum x ↦ sup { p(x), q(x) }. The same is true of the supremum of any nonempty finite family of Fseminorms on X.^{[12]}
 The restriction of an Fseminorm (resp. Fnorm) to a vector subspace is an Fseminorm (resp. Fnorm).^{[9]}
 A nonnegative realvalued function on X is a seminorm if and only if it is a convex Fseminorm, or equivalently, if and only if it is a convex balanced Gseminorm.^{[10]}
 In particular, every seminorm is an Fseminorm.
 For any 0 < p < 1, the map f on ℝ^{n} defined by [f(x_{1}, ..., x_{n})]^{p} := x_{1}^{p} + ⋅⋅⋅ + x_{n}^{p} is an Fnorm that is not a norm.
 If L : X → Y is a linear map and if q is an Fseminorm on Y, then q ∘ L is an Fseminorm on X.^{[12]}
 Let X_{ℂ} be a complex vector space and let X_{ℝ} denote X_{ℂ} considered as a vctor space over ℝ. Any Fseminorm on X_{ℂ} is also an Fseminorm on X_{ℝ}.^{[9]}
Properties of FseminormsEdit
 Every Fseminorm is a paranorm and every paranorm is equivalent to some Fseminorm.^{[7]}
 Every Fseminorm on a vector space X is a value on X. In particular,
 p(0) = 0;
 p(x) = p(x) for all x ∈ X.
Topology induced by a single FseminormEdit
Theorem^{[11]} — Let p be an Fseminorm 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 Fnorm 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 FseminormsEdit
Suppose that 𝒮 is a nonempty collection of Fseminorms on a vector space X and for any finite subset ℱ ⊆ 𝒮 and any r > 0, let
 U_{ℱ, r} := { x ∈ X : p(x) < r }.
The set { U_{ℱ, 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 τ_{𝒮}.^{[12]}
 Each U_{ℱ, r} is a balanced and absorbing subset of X.^{[12]}
 U_{ℱ, r/2} + U_{ℱ, r/2} ⊆ U_{ℱ, r}.^{[12]}
 τ_{𝒮} is the coarsest vector topology on X making each p ∈ 𝒮 continuous.^{[12]}
 τ_{𝒮} is Hausdorff if and only if for every nonzero x ∈ X, there exists some p ∈ 𝒮 such that p(x) > 0.^{[12]}
 If 𝒯 is the set of all continuous Fseminorms on (X, τ_{𝒮}) then τ_{𝒮} = τ_{𝒯}.^{[12]}
 If 𝒯 is the set of all pointwise suprema of nonempty finite subsets of ℱ of 𝒮 then 𝒯 is a directed family of Fseminorms and τ_{𝒮} = τ_{𝒯}.^{[12]}
Fréchet combinationEdit
Suppose that p_{•} = (p_{i})^{∞}
_{i=1} is a family of nonnegative subadditive functions on a vector space X.
Definition:^{[8]} The Fréchet combination of p_{•} is defined to be the realvalued map
 .
As an FseminormEdit
Assume that p_{•} = (p_{i})^{∞}
_{i=1} is an increasing sequence of seminorms on X and let p be the Fréchet combination of p_{•}.
Then p is an Fseminorm on X that induces the same locally convex topology as the family p_{•} of seminorms.^{[13]}
Since p_{•} = (p_{i})^{∞}
_{i=1} is increasing, a basis of open neighborhoods of the origin consists of all sets of the form { x ∈ X : p_{i}(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 Fseminorm 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 paranormEdit
If each p_{i} 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} p_{i}(x) + 1/n : n > 0 is an integer }.^{[8]}  r(x) := ∑^{∞}
_{n=1} min { 1/2^{n}, p_{n}(x) }.^{[8]}
GeneralizationEdit
The Fréchet combination can be generalized by use of a bounded remetrization function.
 Definition:^{[15]} A bounded remetrization function is a continuous nonnegative nondecreasing 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_{•} = (p_{i})^{∞}
_{i=1} is a family of nonnegative Fseminorm on a vector space X, R} is a bounded remetrization function, and r_{•} = (r_{i})^{∞}
_{i=1} is a sequence of positive real numbers whose sum is finite.
Then
defines a bounded Fseminorm that is uniformly equivalent to the p_{•}.^{[16]} It has the property that for any net x_{•} = (x_{i})_{a ∈ A} in X, p (x_{•}) → 0 if and only if p_{i} (x_{•}) → 0 for all i.^{[16]}p is an Fnorm if and only if the p_{•} separate points on X.^{[16]}
CharacterizationsEdit
Of (pseudo)metrics induced by (semi)normsEdit
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 TVSEdit
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]}
 X is pseudometrizable (i.e. the vector topology τ is induced by a pseudometric on X).
 X has a countable neighborhood base at the origin.
 The topology on X is induced by a translationinvariant pseudometric on X.
 The topology on X is induced by an Fseminorm.
 The topology on X is induced by a paranorm.
Of metrizable TVSEdit
If (X, τ) is a TVS then the following are equivalent:
 X is metrizable.
 X is Hausdorff and pseudometrizable.
 X is Hausdorff and has a countable neighborhood base at the origin.^{[11]}^{[12]}
 The topology on X is induced by a translationinvariant metric on X.^{[11]}
 The topology on X is induced by an Fnorm.^{[11]}^{[12]}
 The topology on X is induced by a monotone Fnorm.^{[12]}
 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]}
 The origin { 0 } is closed in X, and there is a countable basis of neighborhoods for 0 in X.
 (X, τ) is metrizable (as a topological space).
 There is a translationinvariant 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 translationinvariant.
Of locally convex pseudometrizable TVSEdit
If (X, τ) is TVS then the following are equivalent:^{[13]}
 X is locally convex and pseudometrizable.
 X has a countable neighborhood base at the origin consisting of convex sets.
 The topology of X is induced by a countable family of (continuous) seminorms.
 The topology of X is induced by a countable increasing sequence of (continuous) seminorms (p_{i})^{∞}
_{i=1} (increasing means that for all i, p_{i} ≤ p_{i+1}).  The topology of X is induced by an Fseminorm of the form:
_{i=1} are (continuous) seminorms on X.^{[18]}
QuotientsEdit
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 Fseminorm on X, then the map P : X/M → ℝ defined by
 P(x + M) := inf { p(x + m) : m ∈ M }
 If in addition p is an Fnorm on X and if M is a closed vector subspace of X then P is an Fnorm on X.^{[11]}
Examples and sufficient conditionsEdit
 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 Fseminorm^{[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 DFspace or an LMspace. If X is a sequential space then it is either metrizable or else a Montel DFspace.
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]}
NormabilityEdit
If X is a Hausdorff locally convex TVS then the following are equivalent:
 X is normable.
 X has a bounded neighborhood of the origin.
 the strong dual of X is normable.^{[23]}
 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 setsEdit
Suppose that (X, d) is a pseudometric space and B ⊆ X. We say that B is metrically bounded or dbounded 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 ddiameter 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 TVSEdit
Theorem^{[25]} — All infinitedimensional 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 nonmeager).^{[27]}
 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 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.^{[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 FrechetUrysohn 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 nonmeager).^{[27]}
 The dimension of a complete metrizable TVS is either finite or uncountable.^{[33]}
CompletenessEdit
Recall that 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^{[34]} — 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.
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 completeTVS.
Theorem^{[37]} — If X is a TVS whose topology is induced by a paranorm p, then X is complete if and only if for every sequence (x_{i})^{∞}
_{i=1} in X, if ∑^{∞}
_{i=1} p(x_{i}) < ∞ then ∑^{∞}
_{i=1} x_{i} converges in X.
 A Baire separable topological group is metrizable if and only if it is cosmic.^{[32]}
 If M is a closed vector subspace of a complete pseudometrizable TVS X, then the quotient space X ∖ M is complete.^{[38]}
 Suppose M is a complete vector subspace of a metrizable TVS X. If the quotient space X ∖ M is complete then so is X.^{[38]}
 Note that if X is not complete then M := X is a closed, but not complete, vector subspace of X.
Subsets and subsequencesEdit
 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 ⊆ cl_{C} R.^{[39]}
 Every totally bounded subset of a locally convex metrizable TVS X is contained in the closed [[Absolutely convex set}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 (x_{i})^{∞}
_{i=1} is a null sequence (i.e. it converges to the origin) in a metrizable TVS then there exists a sequence (r_{i})^{∞}
_{i=1} of positive real numbers diverging to ∞ such that (r_{i}x_{i})^{∞}
_{i=1} → 0.^{[41]}  A subset of a complete metric space is closed if and only if it is complete.
 Note that 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 ⊆ X_{D}, and both X and the auxiliary normed space X_{D} induce the same subspace topology on B.^{[42]}
BanachSaks theorem^{[43]} — If (x_{n})^{∞}
_{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_{•} = (y_{i})^{∞}
_{i=1} in X such that y_{•} → x in (X, 𝜏) and each y_{i} is a convex combination of finitely many x_{n}.
Mackey's countability condition^{[14]} — Suppose that X is a locally convex metrizable TVS and that (B_{i})^{∞}
_{i=1} is a countable sequence of bounded subsets of X.
Then there exists a bounded subset B of X and a sequence (r_{i})^{∞}
_{i=1} of positive real numbers such that B_{i} ⊆ r_{i} B for all i.
Linear mapsEdit
 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 infinitedimensional pseudometrizable TVS.^{[44]}
 Thus, a pseudometrizable TVS is finitedimensional 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:
 F is continuous;
 F is a (locally) bounded map (i.e. F maps (von Neumann) bounded subsets of X to bounded subsets of Y);^{[12]}
 F is sequentially continuous;^{[12]}
 the image under F of every null sequence in X is a bounded set;^{[12]}
 Recall that a null sequence is a sequence that converges to the origin.
 F maps null sequences to null sequences;
 Open and almost open maps
 Theorem:^{[45]} 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.
 Theorem:^{[45]} If T : X → Y is a surjective linear operator from a locally convex space X onto a barrelled space Y then T is almost open. (note that every complete pseudometrizable space is barrelled)
 Theorem:^{[45]} If T : X → Y is a surjective linear operator from a TVS X onto a Baire space Y then T is almost open.
 Theorem:^{[45]} 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 nonmeager in Y then T : X → Y is a surjective open map and Y is a complete metrizable space.
HahnBanach extension propertyEdit
Let X be a TVS. Say that a vector subspace M of 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 X has the HahnBanach extension property (HBEP) if every vector subspace of X has the extension property.^{[22]}
The HahnBanach theorem guarantees that every Hausdorff locally convex space has the HBEP. For complete metrizable TVSs there is a converse:
Theorem^{[22]} (Kalton) — Every complete metrizable TVS with the HahnBanach extension property is locally convex.
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 alsoEdit
 Asymmetric norm – Generalization of the concept of a norm
 Complete metric space – A set with a notion of distance where every sequence of points that get progressively closer to each other will converge
 Complete topological vector space – A TVS where points that get progressively closer to each other will always converge to a point
 Closed graph theorem (functional analysis) – Theorems for deducing continuity from a function's graph
 Equivalence of metrics
 Fspace – Topological vector space with a complete translationinvariant metric
 Fréchet space – A locally convex topological vector space that is also a complete metric space
 Locally convex topological vector space – A vector space with a topology defined by convex open sets
 Metric space – Mathematical set defining distance
 Open mapping theorem (functional analysis) – Theorem giving conditions for a continuous linear map to be an open map
 Pseudometric space – A generalization of metric spaces in which the distance between two distance points can be 0
 Relation of norms and metrics
 Seminorm
 Sublinear function
 Topological vector space – Vector space with a notion of nearness
 Uniform space – Topological space with a notion of uniform properties
 Ursescu theorem
NotesEdit
ReferencesEdit
 ^ Narici & Beckenstein 2011, pp. 118.
 ^ ^{a} ^{b} ^{c} Narici & Beckenstein 2011, pp. 3740.
 ^ ^{a} ^{b} Swartz 1992, p. 15.
 ^ Wilansky 2013, p. 17.
 ^ ^{a} ^{b} Wilansky 2013, pp. 4047.
 ^ Wilansky 2013, p. 15.
 ^ ^{a} ^{b} Schechter 1996, pp. 689691.
 ^ ^{a} ^{b} ^{c} ^{d} ^{e} ^{f} ^{g} ^{h} ^{i} ^{j} ^{k} ^{l} ^{m} ^{n} ^{o} Wilansky 2013, pp. 1518.
 ^ ^{a} ^{b} ^{c} ^{d} Schechter 1996, p. 692.
 ^ ^{a} ^{b} Schechter 1996, p. 691.
 ^ ^{a} ^{b} ^{c} ^{d} ^{e} ^{f} ^{g} ^{h} ^{i} ^{j} ^{k} ^{l} Narici & Beckenstein 2011, pp. 9195.
 ^ ^{a} ^{b} ^{c} ^{d} ^{e} ^{f} ^{g} ^{h} ^{i} ^{j} ^{k} ^{l} ^{m} ^{n} ^{o} ^{p} ^{q} ^{r} ^{s} ^{t} Jarchow 1981, pp. 3842.
 ^ ^{a} ^{b} Narici & Beckenstein 2011, p. 123.
 ^ ^{a} ^{b} ^{c} ^{d} ^{e} ^{f} ^{g} ^{h} Narici & Beckenstein 2011, pp. 156175.
 ^ ^{a} ^{b} ^{c} Schechter 1996, p. 487.
 ^ ^{a} ^{b} ^{c} Schechter 1996, pp. 692693.
 ^ Köthe 1983, section 15.11
 ^ Schechter 1996, p. 706.
 ^ Narici & Beckenstein 2011, pp. 115154.
 ^ Wilansky 2013, pp. 1516.
 ^ Schaefer & Wolff 1999, pp. 9192.
 ^ ^{a} ^{b} ^{c} ^{d} ^{e} Narici & Beckenstein 2011, pp. 225273.
 ^ ^{a} ^{b} Trèves 2006, p. 201.
 ^ Schaefer & Wolff 1999, pp. 6872.
 ^ Wilansky 2013, p. 57.
 ^ Jarchow 1981, p. 222.
 ^ ^{a} ^{b} ^{c} ^{d} Narici & Beckenstein 2011, pp. 371423.
 ^ Schaefer & Wolff 1999, p. 153.
 ^ Narici & Beckenstein 2011, pp. 459483.
 ^ Köthe 1969, p. 168.
 ^ Wilansky 2013, p. 59.
 ^ ^{a} ^{b} Gabriyelyan, S.S. "On topological spaces and topological groups with certain local countable networks (2014)
 ^ ^{a} ^{b} Schaefer & Wolff 1999, pp. 1235.
 ^ Narici & Beckenstein 2011, pp. 4750.
 ^ Schaefer & Wolff 1999, p. 35.
 ^ Klee, V. L. (1952). "Invariant metrics in groups (solution of a problem of Banach)" (PDF). Proc. Amer. Math. Soc. (3): 484–487. doi:10.1090/s00029939195200472504.
 ^ Wilansky 2013, pp. 5657.
 ^ ^{a} ^{b} Narici & Beckenstein 2011, pp. 4766.
 ^ Schaefer & Wolff 1999, pp. 190202.
 ^ Narici & Beckenstein 2011, pp. 172173.
 ^ ^{a} ^{b} Rudin 1991, p. 22.
 ^ Narici & Beckenstein 2011, pp. 441457.
 ^ Rudin 1991, p. 67.
 ^ ^{a} ^{b} Narici & Beckenstein 2011, p. 125.
 ^ ^{a} ^{b} ^{c} ^{d} Narici & Beckenstein 2011, pp. 466468.
BibliographyEdit
 Jarchow, Hans (1981). Locally convex spaces. Stuttgart: B.G. Teubner. ISBN 9783519022244. OCLC 8210342.
 Bourbaki, Nicolas (1987) [1981]. Topological Vector Spaces: Chapters 1–5 [Sur certains espaces vectoriels topologiques]. Annales de l'Institut Fourier. Éléments de mathématique. 2. Translated by Eggleston, H.G.; Madan, S. Berlin New York: SpringerVerlag. ISBN 9783540423386. OCLC 17499190.
 Bourbaki, Nicolas (1950). "Sur certains espaces vectoriels topologiques". Annales de l'Institut Fourier (in French). 2: 5–16 (1951). MR 0042609.
 Edwards, Robert E. (1995). Functional Analysis: Theory and Applications. New York: Dover Publications. ISBN 9780486681436. OCLC 30593138.
 Grothendieck, Alexander (1973). Topological Vector Spaces. Translated by Chaljub, Orlando. New York: Gordon and Breach Science Publishers. ISBN 9780677300207. OCLC 886098.
 Jarchow, Hans (1981). Locally convex spaces. Stuttgart: B.G. Teubner. ISBN 9783519022244. OCLC 8210342.
 Khaleelulla, S. M. (1982). Written at Berlin Heidelberg. Counterexamples in Topological Vector Spaces. Lecture Notes in Mathematics. 936. Berlin New York: SpringerVerlag. ISBN 9783540115656. OCLC 8588370.
 Köthe, Gottfried (1969). Topological Vector Spaces I. Grundlehren der mathematischen Wissenschaften. 159. Translated by Garling, D.J.H. New York: Springer Science & Business Media. ISBN 9783642649882. MR 0248498. OCLC 840293704.
 Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 9781584888666. OCLC 144216834.
 Rudin, Walter (1991). Functional Analysis. International Series in Pure and Applied Mathematics. 8 (Second ed.). New York, NY: McGrawHill Science/Engineering/Math. ISBN 9780070542365. OCLC 21163277.
 Robertson, Alex P.; Robertson, Wendy J. (1980). Topological Vector Spaces. Cambridge Tracts in Mathematics. 53. Cambridge England: Cambridge University Press. ISBN 9780521298827. 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 9781461271550. OCLC 840278135.
 Schechter, Eric (1996). Handbook of Analysis and Its Foundations. San Diego, CA: Academic Press. ISBN 9780126227604. OCLC 175294365.
 Swartz, Charles (1992). An introduction to Functional Analysis. New York: M. Dekker. ISBN 9780824786434. OCLC 24909067.
 Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 9780486453521. OCLC 853623322.
 Wilansky, Albert (2013). Modern Methods in Topological Vector Spaces. Mineola, New York: Dover Publications, Inc. ISBN 9780486493534. OCLC 849801114.
 Husain, Taqdir (1978). Barrelledness in topological and ordered vector spaces. Berlin New York: SpringerVerlag. ISBN 3540090967. OCLC 4493665.