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).

Pseudometrics and metrics

A pseudometric on a set $X$ is a map $d : X \times X \to ℝ$ satisfying the following properties:

$d (x, x) = 0$ for all $x \in X$ ;
Symmetry: $d (x, y) = d (y, x)$ for all $x, y \in X$ ;
Subadditivity: $d (x, z) \leq d (x, y) + d (y, z)$ for all $x, y, z \in X$ .

A pseudometric is called a metric if it satisfies:

Identity of indiscernibles: for all $x, y \in X$ , 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 $x, y \in X$ , $d (x, y) \leq max {d (x, z), d (y, z)$ }.

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 pseudometric

If $d$ is a pseudometric on a set $X$ then collection of open balls:

B r (z) := {x \in 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

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 groups

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 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:

Translation invariance: $d (x + z, y + z) = d (x, y)$ for all $x, y, z \in X$ ;
$d (x, y) = d (x - y, 0)$ for all $x, y \in X$ .

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 $p : X \to ℝ$ with the following properties:[2]

Non-negative: $p \geq 0$ .
Subadditive: $p (x + y) \leq p (x) + p (y)$ for all $x, y \in X$ ;
$p (0) = 0$ .
Symmetric: $p (- x) = p (x)$ for all $x \in X$ .

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

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

Properties of values

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

$| p (x) - p (y) | \leq p (x - y)$ for all $x, y \in X$ .[3]
$p (nx) \leq np (x)$ and $1 / n p (x) \leq p (x / n)$ for all $x \in X$ and positive integers $n$ .[4]
The set ${x \in X : p (x) = 0$ } 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). Convergesly, if $p$ is a value on $X$ then the map ${{{1}}}$ 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:

$τ$ is induced by a pseudometric; (i.e. $(X, τ)$ is pseudometrizable);
$τ$ is induced by a translation-invariant 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 topology

Let $X$ be a non-trivial (i.e. $X \neq { 0$ }) 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 $x, y \in X$ such that $x \neq y$ . 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.

Paranorms

If $X$ is a vector space over the real or complex numbers then a paranorm on $X$ is a G-seminorm (defined above) $p : X \to ℝ$ on $X$ that satisfies any of the following additional conditions, each of which begins with "for all sequences $x • = (x i) \infty i =1$ in $X$ and all convergent sequences of scalars $s • = (s i) \infty i =1$ ":[5]

Continuity of multiplication: if $s$ is a scalar and $x \in X$ are such that $p (x i - x) \to 0$ and $s • \to s$ , then $p (s i x i - sx) \to 0$ .
Both of the conditions:
- if $s • \to 0$ and if $x \in X$ is such that $p (x i - x) \to 0$ then $p (s i x i) \to 0$ ;
- if $p (x i) \to 0$ then $p (s x i) \to 0$ for every scalar $s$ .
Both of the conditions:
- if $p (x i) \to 0$ and $s • \to s$ for some scalar $s$ then $p (s i x i) \to 0$ ;
- if $s • \to 0$ then $p (s i x) \to 0$ for all $x \in X$ .
Separate continuity:[6]
- if $s • \to s$ for some scalar $s$ then $p (s i x - sx) \to 0$ for every $x \in X$ ;
- if $s$ is a scalar, $x \in X$ , and $p (x i - x) \to 0$ then $p (s x i - sx) \to 0$ .

A paranorm is called an 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 $d : X \times X \to ℝ$ defined by $d (x, y) := p (x - y)$ is a translation-invariant pseudometric on $X$ that defines a vector topology on $X$ .[7]

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

the set ${x \in X : p (x) = 0$ } is a vector subspace of $X$ .[7]
$p (x + n) = p (x)$ for all $x, n \in X$ with $p (n) = 0$ .[7]
If a paranorm $p$ satisfies $p (sx) \leq | s | p (x)$ for all $x \in X$ and scalars $s$ , then $p$ is absolutely homogeneity (i.e. equality holds)[7] 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 $τ$ .[7]
If $p$ is a paranorm on $X$ then so is the map $q (x) := p (x)/[1 + p (x)]$ .[7]
Every positive scalar multiple of a paranorm (resp. total paranorm) is again such a paranorm (resp. total paranorm).
Every seminorm is a paranorm.[7]
The restriction of an paranorm (resp. total paranorm) to a vector subspace is an paranorm (resp. total paranorm).[8]
The sum of two paranorms is a paranorm.[7]
If $p$ and $q$ are paranorms on $X$ then so is $(p \land q)(x) := inf {p (y) + q (z) : x = y + z with y, z \in X$ }. Moreover, $p \land q \leq p$ and $p \land q \leq q$ . This makes the set of paranorms on $X$ into a conditionally complete lattice.[7]
Each of the following real-valued maps are paranorms on X := ℝ²:
- $(x, y) \mapsto | x |$
- $(x, y) \mapsto | x | + | y |$
The real-valued map $(x, y) \mapsto \sqrt x 2 + y 2$ is not paranorms on $X := ℝ 2$ .[7]
If $x • = (x i) i \in I$ is a Hamel basis on a vector space $X$ then the real-valued map that sends $x = \sum i \in I s i x i \in X$ (where all but finitely many of the scalars $s i$ are 0) to $\sum i \in I \sqrt | s i |$ is a paranorm on $X$ , which satisfies $p (sx) = \sqrt | s | p (x)$ for all $x \in X$ and scalars $s$ .[7]
The function $p (x) := | sin(π x) | + min {2, | x |$ } is a paranorm on $ℝ$ that is not balanced but nevertheless equivalent to the usual norm on $R$ . Note that the function $x \mapsto | sin(π x) |$ is subadditive.[9]
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 $X ℝ$ .[8]

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 $p : X \to ℝ$ with the following properties:[10]

Non-negative: $p \geq 0$ .
Subadditive: $p (x + y) \leq p (x) + p (y)$ for all $x, y \in X$ ;
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 ${x \in X : p (x) \leq r$ } or ${x \in X : p (x) < r$ } for some $r \geq 0$ is balanced.
for every x ∈ X, p(1/n x) → 0 as n → ∞
- The sequence $(1 / n) \infty n =1$ can be replaced by any positive sequence converging to 0.[11]

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

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

An $F$ -seminorm is called monotone if it satisfies:

Monotone: $p (rx) < p (sx)$ for all non-zero $x \in X$ and all real $s$ and $t$ such that $s < t$ .[11]

F-seminormed spaces

Definition:[11] An $F$ -seminormed space (resp. $F$ -normed space) is a pair

(X, p)

consisting of a vector space

X

and an

F

-seminorm (resp.

F

-norm)

p

on

X

.

Definition:[11] If

(X, p)

and

(Z, q)

are

F

-seminormed spaces then a map

f : X \to Z

is called an isometric embedding if

q (f (x) - f (y)) = p (x - y)

for all

x, y \in X

.

Every isometric embedding of one $F$ -seminormed space into another is a topological embedding, but the converse is not true in general.[11]

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 $x \mapsto sup {p (x), q (x)$ }. The same is true of the supremum of any non-empty finite family of $F$ -seminorms on $X$ .[11]
The restriction of an $F$ -seminorm (resp. $F$ -norm) to a vector subspace is an $F$ -seminorm (resp. $F$ -norm).[8]
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.[9]
- In particular, every seminorm is an $F$ -seminorm.
For any $0 < p < 1$ , the map $f$ on $ℝ n$ defined by $[f (x 1, ..., x n)] p := | x 1 | p + \cdot\cdot\cdot + | x n | p$ is an $F$ -norm that is not a norm.
If $L : X \to Y$ is a linear map and if $q$ is an $F$ -seminorm on $Y$ , then $q \circ L$ is an $F$ -seminorm on $X$ .[11]
Let $X ℂ$ be a complex vector space and let $X ℝ$ denote $X ℂ$ considered as a vctor space over $ℝ$ . Any $F$ -seminorm on $X ℂ$ is also an $F$ -seminorm on $X ℝ$ .[8]

Properties of F-seminorms

Every $F$ -seminorm is a paranorm and every paranorm is equivalent to some $F$ -seminorm.[12]
Every F-seminorm on a vector space X is a value on X. In particular,
- $p (0) = 0$ ;
- $p (x) = p (- x)$ for all $x \in X$ .

Topology induced by a single F-seminorm

Theorem[10] — Let $p$ be an $F$ -seminorm on a vector space $X$ . Then the map $d : X \times X \to ℝ$ 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 ${x X : p (x) \leq r$ }, 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 $ℱ \subseteq 𝒮$ and any $r > 0$ , let

U ℱ, r := \cap p \in ℱ {x \in X : p (x) < r

}.

The set ${U ℱ, r : r > 0, ℱ \subseteq 𝒮, ℱ finite$ } forms a filter base on $X$ that also forms a neighborhood basis at the origin for a vector topology on $X$ denoted by $τ 𝒮$ .[11]

Each $U ℱ, r$ is a balanced and absorbing subset of $X$ .[11]
$U ℱ, r /2 + U ℱ, r /2 \subseteq U ℱ, r$ .[11]
$τ 𝒮$ is the coarsest vector topology on $X$ making each $p \in 𝒮$ continuous.[11]
$τ 𝒮$ is Hausdorff if and only if for every non-zero $x \in X$ , there exists some $p \in 𝒮$ such that $p (x) > 0$ .[11]
If $𝒯$ is the set of all continuous $F$ -seminorms on $(X, τ 𝒮)$ then $τ 𝒮 = τ 𝒯$ .[11]
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 $τ 𝒮 = τ 𝒯$ .[11]

Fréchet combination

Suppose that $p • = (p i) \infty i =1$ is a family of non-negative subadditive functions on a vector space $X$ .

Definition:[7] The Fréchet combination of $p •$ is defined to be the real-valued map

p(x):=\sum _{i=1}^{\infty }{\frac {p_{i}(x)}{2^{i}\left[1+p_{i}(x)\right]}}

.

As an F-seminorm

Assume that $p • = (p i) \infty i =1$ 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 $p • = (p i) \infty i =1$ is increasing, a basis of open neighborhoods of the origin consists of all sets of the form ${x \in X : p i (x) < r$ } 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 $F$ -seminorm $p$ is

d(x,y)=\sum _{i=1}^{\infty }{\frac {1}{2^{i}}}{\frac {p_{i}(x-y)}{1+p_{i}(x-y)}}

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

As a paranorm

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.[7] This is also true of the following paranorms on $X$ :

$q (x) := inf {\sum n i =1 p i (x) + 1 / n : n > 0 is an integer$ }.[7]
$r (x) := \sum \infty n =1 min {1 / 2 n, p n (x)$ }.[7]

Generalization

The Fréchet combination can be generalized by use of a bounded remetrization function.

Definition:[15] A bounded remetrization function is a continuous non-negative non-decreasing map

R : [0, \infty) \to [0, \infty)

that is subadditive (i.e.

R (s + t) \leq R (s) + R (t)

for all

s, t \geq 0

), has a bounded range, and satisfies

R (s) = 0

if and only if

s = 0

.

Examples of bounded remetrization functions include $arctan t$ , $tanh t$ , $t \mapsto min {t, 1$ }, and $t \mapsto t / 1 + t$ .[15] If $d$ is a pseudometric (resp. metric) on $X$ and $R$ } is a bounded remetrization function then $R \circ d$ is a bounded pseudometric (resp. bounded metric) on $X$ that is uniformly equivalent to $d$ .[15]

Suppose that $p • = (p i) \infty i =1$ is a family of non-negative $F$ -seminorm on a vector space $X$ , $R$ } is a bounded remetrization function, and $r • = (r i) \infty i =1$ is a sequence of positive real numbers whose sum is finite. Then

p(x):=\sum _{i=1}^{\infty }r_{i}R(p_{i}(x))

defines a bounded $F$ -seminorm that is uniformly equivalent to the $p •$ .[16] It has the property that for any net $x • = (x i) a \in A$ in $X$ , $p (x •) \to 0$ if and only if $p i (x •) \to 0$ for all $i$ .[16] $p$ is an $F$ -norm if and only if the $p •$ separate points on $X$ .[16]

Characterizations

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:[10]

$X$ 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 translation-invariant pseudometric on $X$ .
The topology on $X$ is induced by an $F$ -seminorm.
The topology on $X$ is induced by a paranorm.

Of metrizable TVS

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.[10][11]
The topology on $X$ is induced by a translation-invariant metric on $X$ .[10]
The topology on $X$ is induced by an $F$ -norm.[10][11]
The topology on $X$ is induced by a monotone $F$ -norm.[11]
The topology on $X$ is induced by a total paranorm.

Of locally convex pseudometrizable TVS

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 $(p i) \infty i =1$ (increasing means that for all $i$ , $p i \leq p i +1$ ).
The topology of $X$ is induced by an $F$ -seminorm of the form:
$p(x)=\sum _{n=1}^{\infty }2^{-n}\operatorname {arctan} p_{n}(x)$
where $(p i) \infty i =1$ are (continuous) seminorms on $X$ .[17]

Quotients

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

If $X$ is a pseudometrizable TVS then so is $X / M$ .[10]
If $X$ is a complete pseudometrizable TVS and $M$ is a closed vector subspace of $X$ then $X / M$ is complete.[10]
If $X$ is metrizable TVS and $M$ is a closed vector subspace of $X$ then $X / M$ is metrizable.[10]
If p is an F-seminorm on X, then the map P : X/M → ℝ defined by
$P (x + M) := inf {p (x + m) : m \in M$ }

is an $F$ -seminorm on $X / M$ that induces the usual quotient topology on $X / M$ .[10]
- 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 $X$ .[10]

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 $x, y \in X$ .[18].
If (X, d) is pseudometric TVS with a translation invariant pseudometric d, then p(x) := d(x, 0) defines a paranorm.[19]
- 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[20] 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]
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]

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 $X_{b}^{\prime }$ of $X$ is normable.[21]
the strong dual $X_{b}^{\prime }$ of $X$ is metrizable.[21]

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]

Metrically bounded sets and bounded sets

Suppose that $(X, d)$ is a pseudometric space and $B \subseteq X$ . We say that $B$ is metrically bounded or $d$ -bounded if there exists a real number $R > 0$ such that $d (x, y) \leq R$ for all $x, y \in B$ ; 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[23] — All infinite-dimensional separable complete metrizable TVS are homeomorphic.

Every metrizable locally convex TVS is a quasibarrelled space,[24] bornological space, and a Mackey space.

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.[25]

If $X$ is a complete pseudo-metrizable TVS and $M$ is a closed vector subspace of $X$ , then $X / M$ is complete.[10]

The strong dual of a locally convex metrizable TVS is a webbed space.[26]

If $(X, 𝜏)$ and $(X, 𝜐)$ are complete metrizable TVSs and if $𝜐$ is coarser than $𝜏$ then $𝜏 = 𝜐$ .[27] This is no longer true if either one of these metrizable TVSs is not complete.[28]

Completeness

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[29] — 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[30] — If $X$ is a TVS whose topology is induced by a paranorm $p$ , then $X$ is complete if and only if for every sequence $(x i) \infty i =1$ in $X$ , if $\sum \infty i =1 p (x i) < \infty$ then $\sum \infty i =1 x i$ converges in $X$ .

Subsets and subsequences

In a pseudometrizable TVS, every bornivore is a neighborhood of the origin.[31]
If $d$ is a translation invariant metric on a vector space $X$ , then $d (nx, 0) \leq nd (x, 0)$ for all $x \in X$ and every positive integer $n$ .[32]
If $(x i) \infty i =1$ is a null sequence (i.e. it converges to the origin) in a metrizable TVS then there exists a sequence $(r i) \infty i =1$ of positive real numbers diverging to $\infty$ such that $(r i x i) \infty i =1 \to 0$ .[32]

Banach-Saks theorem[33] — If $(x n) \infty n =1$ is a sequence in a locally convex metrizable TVS $(X, 𝜏)$ that converges weakly to some $x \in X$ , then there exists a sequence $y • = (y i) \infty i =1$ in $X$ such that $y • \to x$ 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 $(B i) \infty i =1$ is a countable sequence of bounded subsets of $X$ . Then there exists a bounded subset $B$ of $X$ and a sequence $(r i) \infty i =1$ of positive real numbers such that $B i \subseteq r i B$ 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.[34]
- Thus, a pseudometrizable TVS is finite-dimensional if and only if its continuous dual space is equal to its algebraic dual space.[34]

If $F : X \to Y$ 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 $Y$ );[11]
$F$ is sequentially continuous;[11]
the image under F of every null sequence in X is a bounded set;[11]
- Recall that a null sequence is a sequence that converges to the origin.
$F$ maps null sequences to null sequences;

Hahn-Banach extension property

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 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[22] (Kalton) — Every complete metrizable TVS with the Hahn-Banach 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]

References

Narici 2011, pp. 1-18.
Narici 2011, pp. 37-40.
Swartz 1992, p. 15.
Wilansky 2013, p. 17.
Wilansky 2013, p. 15.
Schechter pp.689-691
Wilansky 2013, pp. 15-18.
Schechter 1996, p. 692.
Schechter 1996, p. 691.
Narici 2011, pp. 91-95.
Jarchow 1981, pp. 38-42.
Schechter pp. 689-691
Narici 2011, p. 123.
Narici 2011, pp. 156-175.
Schechter 1996, p. 487.
Schechter 1996, pp. 692-693.
Schechter p. 706
Narici 2011, pp. 115-154.
Wilansky 2013, pp. 15-16.
Schaefer 1999, pp. 91-92.
Treves 2006, p. 201.
Narici 2011, pp. 225-273.
Wilansky 2013, p. 57.
Jarchow 1981, p. 222.
Schaefer & Wolff 1999, p. 153.
Narici 2011, pp. 459-483.
Köthe 1969, p. 168.
Wilansky 2013, p. 59.
Narici 2011, pp. 47-50.
Wilansky 2013, pp. 56-57.
Narici 2011, pp. 172-173.
Rudin 1991, p. 22.
Rudin 1991, p. 67.
Narici 2011, p. 125.

Jarchow, Hans (1981). Locally convex spaces. Stuttgart: B.G. Teubner. ISBN 978-3-519-02224-4. OCLC 8210342.CS1 maint: ref=harv (link)
Bourbaki, Nicolas (1987) [1981]. Topological Vector Spaces: Chapters 1–5 [Sur certains espaces vectoriels topologiques]. Annales de l'Institut Fourier. Elements of mathematics (in French). 2. Translated by Eggleston, H.G.; Madan, S. Berlin New York: Springer-Verlag. ISBN 978-3-540-42338-6. OCLC 17499190.CS1 maint: ref=harv (link)
Bourbaki, Nicolas (1950). "Sur certains espaces vectoriels topologiques". Annales de l'Institut Fourier (in French). 2: 5–16 (1951). MR 0042609.
Edwards, Robert E. (Jan 1, 1995). Functional Analysis: Theory and Applications. New York: Dover Publications. ISBN 978-0-486-68143-6. OCLC 30593138.CS1 maint: ref=harv (link) CS1 maint: date and year (link)
Grothendieck, Alexander (January 1, 1973). Topological Vector Spaces. Translated by Chaljub, Orlando. New York: Gordon and Breach Science Publishers. ISBN 978-0-677-30020-7. OCLC 886098.CS1 maint: ref=harv (link) CS1 maint: date and year (link)
Jarchow, Hans (1981). Locally convex spaces. Stuttgart: B.G. Teubner. ISBN 978-3-519-02224-4. OCLC 8210342.CS1 maint: ref=harv (link)
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.CS1 maint: ref=harv (link) CS1 maint: date and year (link)
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 978-3-642-64988-2. MR 0248498. OCLC 840293704.CS1 maint: ref=harv (link)
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.CS1 maint: ref=harv (link) CS1 maint: date and year (link)
Robertson, Alex P.; Robertson, Wendy J. (January 1, 1980). Topological Vector Spaces. Cambridge Tracts in Mathematics. 53. Cambridge England: Cambridge University Press. ISBN 978-0-521-29882-7. OCLC 589250.CS1 maint: ref=harv (link) CS1 maint: date and year (link)
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.CS1 maint: ref=harv (link)
Swartz, Charles (1992). An introduction to Functional Analysis. New York: M. Dekker. ISBN 978-0-8247-8643-4. OCLC 24909067.CS1 maint: ref=harv (link)
Trèves, François (August 6, 2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.CS1 maint: ref=harv (link) CS1 maint: date and year (link)
Wilansky, Albert (2013). Modern Methods in Topological Vector Spaces. Mineola, New York: Dover Publications, Inc. ISBN 978-0-486-49353-4. OCLC 849801114.CS1 maint: ref=harv (link)
Husain, Taqdir (1978). Barrelledness in topological and ordered vector spaces. Berlin New York: Springer-Verlag. ISBN 3-540-09096-7. OCLC 4493665.CS1 maint: ref=harv (link)

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[FOOTNOTENarici20111-18-1] Narici 2011, pp. 1-18.

[FOOTNOTENarici201137-40-2] Narici 2011, pp. 37-40.

[FOOTNOTESwartz199215-3] Swartz 1992, p. 15.

[FOOTNOTEWilansky201317-4] Wilansky 2013, p. 17.

[FOOTNOTEWilansky201315-5] Wilansky 2013, p. 15.

[6] Schechter pp.689-691

[FOOTNOTEWilansky201315-18-7] Wilansky 2013, pp. 15-18.

[FOOTNOTESchechter1996692-8] Schechter 1996, p. 692.

[FOOTNOTESchechter1996691-9] Schechter 1996, p. 691.

[FOOTNOTENarici201191-95-10] Narici 2011, pp. 91-95.

[FOOTNOTEJarchow198138-42-11] Jarchow 1981, pp. 38-42.

[12] Schechter pp. 689-691

[FOOTNOTENarici2011123-13] Narici 2011, p. 123.

[FOOTNOTENarici2011156-175-14] Narici 2011, pp. 156-175.

[FOOTNOTESchechter1996487-15] Schechter 1996, p. 487.

[FOOTNOTESchechter1996692-693-16] Schechter 1996, pp. 692-693.

[17] Schechter p. 706

[FOOTNOTENarici2011115-154-18] Narici 2011, pp. 115-154.

[FOOTNOTEWilansky201315-16-19] Wilansky 2013, pp. 15-16.

[FOOTNOTESchaefer199991-92-20] Schaefer 1999, pp. 91-92.

[FOOTNOTETreves2006201-21] Treves 2006, p. 201.

[FOOTNOTENarici2011225-273-22] Narici 2011, pp. 225-273.

[FOOTNOTEWilansky201357-23] Wilansky 2013, p. 57.

[FOOTNOTEJarchow1981222-24] Jarchow 1981, p. 222.

[FOOTNOTESchaeferWolff1999153-25] Schaefer & Wolff 1999, p. 153.

[FOOTNOTENarici2011459-483-26] Narici 2011, pp. 459-483.

[FOOTNOTEKöthe1969168-27] Köthe 1969, p. 168.

[FOOTNOTEWilansky201359-28] Wilansky 2013, p. 59.

[FOOTNOTENarici201147-50-29] Narici 2011, pp. 47-50.

[FOOTNOTEWilansky201356-57-30] Wilansky 2013, pp. 56-57.

[FOOTNOTENarici2011172-173-31] Narici 2011, pp. 172-173.

[FOOTNOTERudin199122-32] Rudin 1991, p. 22.

[FOOTNOTERudin199167-33] Rudin 1991, p. 67.

[FOOTNOTENarici2011125-34] Narici 2011, p. 125.

Functional analysis (topics, glossary)
Spaces	Hilbert space, Banach space, Fréchet space, topological vector space
Theorems	Hahn–Banach theorem, closed graph theorem, uniform boundedness principle, Kakutani fixed-point theorem, Krein–Milman theorem, min-max theorem, Gelfand–Naimark theorem, Banach–Alaoglu theorem
Operators	bounded operator, compact operator, adjoint operator, unitary operator, Hilbert–Schmidt operator, trace class, unbounded operator
Algebras	Banach algebra, C-algebra, spectrum of a C-algebra, operator algebra, group algebra of a locally compact group, von Neumann algebra
Open problems	invariant subspace problem, Mahler's conjecture
Applications	Besov space, Hardy space, spectral theory of ordinary differential equations, heat kernel, index theorem, calculus of variation, functional calculus, integral operator, Jones polynomial, topological quantum field theory, noncommutative geometry, Riemann hypothesis
Advanced topics	locally convex space, approximation property, balanced set, Schwartz space, weak topology, barrelled space, Banach–Mazur distance, Tomita–Takesaki theory

Topological vector spaces (TVSs)
Basic concepts	Banach space Continuous linear operator Functionals Hilbert space Linear operators Locally convex space Homomorphism Topological vector space Vector space
Main results	Closed graph theorem F. Riesz's theorem Hahn–Banach (hyperplane separation Vector-Valued Hahn-Banach) Open mapping (Banach–Schauder) (Bounded inverse) Uniform boundedness (Banach–Steinhaus)
Maps	Almost open Bilinear (form operator) and Sesquilinear forms Closed Compact operator Continuous and Discontinuous Linear maps Densely defined Homomorphism Functionals Norm Operator Seminorm Sublinear Transpose
Types of sets	Absolutely convex/disk Absorbing/Radial Affine Balanced/Circled Bounding points Bounded Complemented subspace Convex Convex cone (subset) Linear cone (subset) Extreme point Pre-compact/Totally bounded Radial Radially convex/Star-shaped Symmetric
Set operations	Affine hull (Relative) Algebraic interior (core) Convex hull Linear span Minkowski addition Polar (Quasi) Relative interior
Types of TVSs	Asplund B-complete/Ptak Banach (Countably) Barrelled (Ultra-) Bornological Brauner Complete (DF)-space Distinguished F-space Fréchet (tame Fréchet) Grothendieck Hilbert Infrabarreled Interpolation space LB-space LF-space Locally convex space Mackey (Pseudo)Metrizable Montel Quasibarrelled Quasi-complete Quasinormed (Polynomially Semi-) Reflexive Riesz Schwartz Semi-complete Smith Stereotype (B Strictly Uniformly convex (Quasi-) Ultrabarrelled Uniformly smooth Webbed With the Approximation property