Metrizable topological vector space

In functional analysis and related areas of mathematics, a metrizable (resp. pseudometrizable) topological vector space (TVS) is a TVS whose topology is induced by a metric (resp. pseudometric). An LM-space is an inductive limit of a sequence of locally convex metrizable TVS.

Pseudometrics and metrics

A pseudometric on a set ${\displaystyle X}$ is a map ${\displaystyle d:X\times X\rightarrow \mathbb {R} }$ satisfying the following properties:

1. ${\displaystyle d(x,x)=0{\text{ for all }}x\in X}$;
2. Symmetry: ${\displaystyle d(x,y)=d(y,x){\text{ for all }}x,y\in X}$;
3. Subadditivity: ${\displaystyle d(x,z)\leq d(x,y)+d(y,z){\text{ for all }}x,y,z\in X.}$

A pseudometric is called a metric if it satisfies:

4. Identity of indiscernibles: for all ${\displaystyle x,y\in X,}$ if ${\displaystyle d(x,y)=0}$ then ${\displaystyle x=y.}$

Ultrapseudometric

A pseudometric ${\displaystyle d}$ on ${\displaystyle X}$ is called a ultrapseudometric or a strong pseudometric if it satisfies:

5. Strong/Ultrametric triangle inequality: ${\displaystyle d(x,z)\leq \max\{d(x,y),d(y,z)\}{\text{ for all }}x,y,z\in X.}$

Pseudometric space

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

Topology induced by a pseudometric

If ${\displaystyle d}$ is a pseudometric on a set ${\displaystyle X}$ then collection of open balls:

as ${\displaystyle z}$ ranges over ${\displaystyle X}$ and ${\displaystyle r>0}$ ranges over the positive real numbers, forms a basis for a topology on ${\displaystyle X}$ that is called the ${\displaystyle d}$-topology or the pseudometric topology on ${\displaystyle X}$ induced by ${\displaystyle d.}$

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

Pseudometrizable space

A topological space ${\displaystyle (X,\tau )}$ is called pseudometrizable (resp. metrizable, ultrapseudometrizable) if there exists a pseudometric (resp. metric, ultrapseudometric) ${\displaystyle d}$ on ${\displaystyle X}$ such that ${\displaystyle \tau }$ is equal to the topology induced by ${\displaystyle d.}$^[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 ${\displaystyle \tau }$ on a real or complex vector space ${\displaystyle X}$ is called a vector topology or a TVS topology if it makes the operations of vector addition and scalar multiplication continuous (that is, if it makes ${\displaystyle X}$ into a topological vector space).

Every topological vector space (TVS) ${\displaystyle X}$ is an additive commutative topological group but not all group topologies on ${\displaystyle X}$ are vector topologies. This is because despite it making addition and negation continuous, a group topology on a vector space ${\displaystyle X}$ may fail to make scalar multiplication continuous. For instance, the discrete topology on any non-trivial vector space makes addition and negation continuous but do not make scalar multiplication continuous.

Translation invariant pseudometrics

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

1. Translation invariance: ${\displaystyle d(x+z,y+z)=d(x,y){\text{ for all }}x,y,z\in X}$;
2. ${\displaystyle d(x,y)=d(x-y,0){\text{ for all }}x,y\in X.}$

Value/G-seminorm

If ${\displaystyle X}$ is a topological group the a value or G-seminorm on ${\displaystyle X}$ (the G stands for Group) is a real-valued map ${\displaystyle p:X\rightarrow \mathbb {R} }$ with the following properties:^[2]

1. Non-negative: ${\displaystyle p\geq 0.}$
2. Subadditive: ${\displaystyle p(x+y)\leq p(x)+p(y){\text{ for all }}x,y\in X}$;
3. ${\displaystyle p(0)=0..}$
4. Symmetric: ${\displaystyle p(-x)=p(x){\text{ for all }}x\in X.}$

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

5. Total/Positive definite: If ${\displaystyle p(x)=0}$ then ${\displaystyle x=0.}$

Properties of values

If ${\displaystyle p}$ is a value on a vector space ${\displaystyle X}$ then:

• ${\displaystyle |p(x)-p(y)|\leq p(x-y){\text{ for all }}x,y\in X.}$^[3]
• ${\displaystyle p(nx)\leq np(x)}$ and ${\displaystyle {\frac {1}{n}}p(x)\leq p(x/n)}$ for all ${\displaystyle x\in X}$ and positive integers ${\displaystyle n.}$^[4]
• The set ${\displaystyle \{x\in X:p(x)=0\}}$ is an additive subgroup of ${\displaystyle X.}$^[3]

Equivalence on topological groups

Pseudometrizable topological groups

An invariant pseudometric that doesn't induce a vector topology

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

This example shows that a translation-invariant (pseudo)metric is not enough to guarantee a vector topology, which leads us to define paranorms and F-seminorms.

Additive sequences

A collection ${\displaystyle {\mathcal {N}}}$ of subsets of a vector space is called additive^[5] if for every ${\displaystyle N\in {\mathcal {N}},}$ there exists some ${\displaystyle U\in {\mathcal {N}}}$ such that ${\displaystyle U+U\subseteq N.}$

All of the above conditions are consequently a necessary for a topology to form a vector topology. Additive sequences of sets have the particularly nice property that they define non-negative continuous real-valued subadditive functions. These functions can then be used to prove many of the basic properties of topological vector spaces and also show that a Hausdorff TVS with a countable basis of neighborhoods is metrizable. The following theorem is true more generally for commutative additive topological groups.

Paranorms

If ${\displaystyle X}$ is a vector space over the real or complex numbers then a paranorm on ${\displaystyle X}$ is a G-seminorm (defined above) ${\displaystyle p:X\rightarrow \mathbb {R} }$ on ${\displaystyle X}$ that satisfies any of the following additional conditions, each of which begins with "for all sequences ${\displaystyle x_{\bullet }=\left(x_{i}\right)_{i=1}^{\infty }}$ in ${\displaystyle X}$ and all convergent sequences of scalars ${\displaystyle s_{\bullet }=\left(s_{i}\right)_{i=1}^{\infty }}$":^[6]

1. Continuity of multiplication: if ${\displaystyle s}$ is a scalar and ${\displaystyle x\in X}$ are such that ${\displaystyle p\left(x_{i}-x\right)\to 0}$ and ${\displaystyle s_{\bullet }\to s,}$ then ${\displaystyle p\left(s_{i}x_{i}-sx\right)\to 0.}$
2. Both of the conditions:
   • if ${\displaystyle s_{\bullet }\to 0}$ and if ${\displaystyle x\in X}$ is such that ${\displaystyle p\left(x_{i}-x\right)\to 0}$ then ${\displaystyle p\left(s_{i}x_{i}\right)\to 0}$;
   • if ${\displaystyle p\left(x_{\bullet }\right)\to 0}$ then ${\displaystyle p\left(sx_{i}\right)\to 0}$ for every scalar ${\displaystyle s.}$
3. Both of the conditions:
   • if ${\displaystyle p\left(x_{\bullet }\right)\to 0}$ and ${\displaystyle s_{\bullet }\to s}$ for some scalar ${\displaystyle s}$ then ${\displaystyle p\left(s_{i}x_{i}\right)\to 0}$;
   • if ${\displaystyle s_{\bullet }\to 0}$ then ${\displaystyle p\left(s_{i}x\right)\to 0{\text{ for all }}x\in X.}$
4. Separate continuity:^[7]
   • if ${\displaystyle s_{\bullet }\to s}$ for some scalar ${\displaystyle s}$ then ${\displaystyle p\left(sx_{i}-sx\right)\to 0}$ for every ${\displaystyle x\in X}$;
   • if ${\displaystyle s}$ is a scalar, ${\displaystyle x\in X,}$ and ${\displaystyle p\left(x_{i}-x\right)\to 0}$ then ${\displaystyle p\left(sx_{i}-sx\right)\to 0}$ .

A paranorm is called total if in addition it satisfies:

• Total/Positive definite: ${\displaystyle p(x)=0}$ implies ${\displaystyle x=0.}$

Properties of paranorms

If ${\displaystyle p}$ is a paranorm on a vector space ${\displaystyle X}$ then the map ${\displaystyle d:X\times X\rightarrow \mathbb {R} }$ defined by ${\displaystyle d(x,y):=p(x-y)}$ is a translation-invariant pseudometric on ${\displaystyle X}$ that defines a vector topology on ${\displaystyle X.}$^[8]

If ${\displaystyle p}$ is a paranorm on a vector space ${\displaystyle X}$ then:

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

Examples of paranorms

• If ${\displaystyle d}$ is a translation-invariant pseudometric on a vector space ${\displaystyle X}$ that induces a vector topology ${\displaystyle \tau }$ on ${\displaystyle X}$ (i.e. ${\displaystyle (X,\tau )}$ is a TVS) then the map ${\displaystyle p(x):=d(x-y,0)}$ defines a continuous paranorm on ${\displaystyle (X,\tau )}$; moreover, the topology that this paranorm ${\displaystyle p}$ defines in ${\displaystyle X}$ is ${\displaystyle \tau .}$^[8]
• If ${\displaystyle p}$ is a paranorm on ${\displaystyle X}$ then so is the map ${\displaystyle q(x):=p(x)/[1+p(x)].}$^[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 ${\displaystyle p}$ and ${\displaystyle q}$ are paranorms on ${\displaystyle X}$ then so is ${\displaystyle (p\wedge q)(x):=\inf _{}\{p(y)+q(z):x=y+z{\text{ with }}y,z\in X\}.}$ Moreover, ${\displaystyle (p\wedge q)\leq p}$ and ${\displaystyle (p\wedge q)\leq q.}$ This makes the set of paranorms on ${\displaystyle X}$ into a conditionally complete lattice.^[8]
• Each of the following real-valued maps are paranorms on ${\displaystyle X:=\mathbb {R} ^{2}}$:
  • ${\displaystyle (x,y)\mapsto |x|}$
  • ${\displaystyle (x,y)\mapsto |x|+|y|}$
• The real-valued maps ${\displaystyle (x,y)\mapsto {\sqrt {\left|x^{2}-y^{2}\right|}}}$ and ${\displaystyle (x,y)\mapsto \left|x^{2}-y^{2}\right|^{3/2}}$ are not a paranorms on ${\displaystyle X:=\mathbb {R} ^{2}.}$^[8]
• If ${\displaystyle x_{\bullet }=\left(x_{i}\right)_{i\in I}}$ is a Hamel basis on a vector space ${\displaystyle X}$ then the real-valued map that sends ${\displaystyle x=\sum _{i\in I}s_{i}x_{i}\in X}$ (where all but finitely many of the scalars ${\displaystyle s_{i}}$ are 0) to ${\displaystyle \sum _{i\in I}{\sqrt {\left|s_{i}\right|}}}$ is a paranorm on ${\displaystyle X,}$ which satisfies ${\displaystyle p(sx)={\sqrt {|s|}}p(x)}$ for all ${\displaystyle x\in X}$ and scalars ${\displaystyle s.}$^[8]
• The function ${\displaystyle p(x):=|\sin(\pi x)|+\min\{2,|x|\}}$is a paranorm on ${\displaystyle \mathbb {R} }$ that is not balanced but nevertheless equivalent to the usual norm on ${\displaystyle R.}$ Note that the function ${\displaystyle x\mapsto |\sin(\pi x)|}$ is subadditive.^[10]
• Let ${\displaystyle X_{\mathbb {C} }}$ be a complex vector space and let ${\displaystyle X_{\mathbb {R} }}$ denote ${\displaystyle X_{\mathbb {C} }}$ considered as a vector space over ${\displaystyle \mathbb {R} .}$ Any paranorm on ${\displaystyle X_{\mathbb {C} }}$ is also a paranorm on ${\displaystyle X_{\mathbb {R} }.}$^[9]

F-seminorms

If ${\displaystyle X}$ is a vector space over the real or complex numbers then an F-seminorm on ${\displaystyle X}$ (the ${\displaystyle F}$ stands for Fréchet) is a real-valued map ${\displaystyle p:X\to \mathbb {R} }$ with the following four properties: ^[11]

1. Non-negative: ${\displaystyle p\geq 0.}$
2. Subadditive: ${\displaystyle p(x+y)\leq p(x)+p(y)}$ for all ${\displaystyle x,y\in X}$
3. Balanced: ${\displaystyle p(ax)\leq p(x)}$ for ${\displaystyle x\in X}$ all scalars ${\displaystyle a}$ satisfying ${\displaystyle |a|\leq 1;}$
   • This condition guarantees that each set of the form ${\displaystyle \{z\in X:p(z)\leq r\}}$ or ${\displaystyle \{z\in X:p(z)<r\}}$ for some ${\displaystyle r\geq 0}$ is a balanced set.
4. For every ${\displaystyle x\in X,}$ ${\displaystyle p\left({\tfrac {1}{n}}x\right)\to 0}$ as ${\displaystyle n\to \infty }$
   • The sequence ${\displaystyle \left({\tfrac {1}{n}}\right)_{n=1}^{\infty }}$ can be replaced by any positive sequence converging to the zero.^[12]

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

5. Total/Positive definite: ${\displaystyle p(x)=0}$ implies ${\displaystyle x=0.}$

An F-seminorm is called monotone if it satisfies:

6. Monotone: ${\displaystyle p(rx)<p(sx)}$ for all non-zero ${\displaystyle x\in X}$ and all real ${\displaystyle s}$ and ${\displaystyle t}$ such that ${\displaystyle s<t.}$^[12]

F-seminormed spaces

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

If ${\displaystyle (X,p)}$ and ${\displaystyle (Z,q)}$ are F-seminormed spaces then a map ${\displaystyle f:X\to Z}$ is called an isometric embedding^[12] if ${\displaystyle q(f(x)-f(y))=p(x,y){\text{ for all }}x,y\in X.}$

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

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 ${\displaystyle p}$ and ${\displaystyle q}$ are F-seminorms on ${\displaystyle X}$ then so is their pointwise supremum ${\displaystyle x\mapsto \sup\{p(x),q(x)\}.}$ The same is true of the supremum of any non-empty finite family of F-seminorms on ${\displaystyle X.}$^[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 ${\displaystyle X}$ 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 ${\displaystyle 0<p<1,}$ the map ${\displaystyle f}$ on ${\displaystyle \mathbb {R} ^{n}}$ defined by
  $${\displaystyle [f\left(x_{1},\ldots ,x_{n}\right)]^{p}=\left|x_{1}\right|^{p}+\cdots \left|x_{n}\right|^{p}}$$

  
  is an F-norm that is not a norm.
• If ${\displaystyle L:X\to Y}$ is a linear map and if ${\displaystyle q}$ is an F-seminorm on ${\displaystyle Y,}$ then ${\displaystyle q\circ L}$ is an F-seminorm on ${\displaystyle X.}$^[12]
• Let ${\displaystyle X_{\mathbb {C} }}$ be a complex vector space and let ${\displaystyle X_{\mathbb {R} }}$ denote ${\displaystyle X_{\mathbb {C} }}$ considered as a vector space over ${\displaystyle \mathbb {R} .}$ Any F-seminorm on ${\displaystyle X_{\mathbb {C} }}$ is also an F-seminorm on ${\displaystyle X_{\mathbb {R} }.}$^[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 ${\displaystyle X}$ is a value on ${\displaystyle X.}$ In particular, ${\displaystyle p(x)=0,}$ and ${\displaystyle p(x)=p(-x)}$ for all ${\displaystyle x\in X.}$

Topology induced by a single F-seminorm

Topology induced by a family of F-seminorms

Suppose that ${\displaystyle {\mathcal {L}}}$ is a non-empty collection of F-seminorms on a vector space ${\displaystyle X}$ and for any finite subset ${\displaystyle {\mathcal {F}}\subseteq {\mathcal {L}}}$ and any ${\displaystyle r>0,}$ let

The set ${\displaystyle \left\{U_{{\mathcal {F}},r}~:~r>0,{\mathcal {F}}\subseteq {\mathcal {L}},{\mathcal {F}}{\text{ finite }}\right\}}$ forms a filter base on ${\displaystyle X}$ that also forms a neighborhood basis at the origin for a vector topology on ${\displaystyle X}$ denoted by ${\displaystyle \tau _{\mathcal {L}}.}$^[12] Each ${\displaystyle U_{{\mathcal {F}},r}}$ is a balanced and absorbing subset of ${\displaystyle X.}$^[12] These sets satisfy^[12]

• ${\displaystyle \tau _{\mathcal {L}}}$ is the coarsest vector topology on ${\displaystyle X}$ making each ${\displaystyle p\in {\mathcal {L}}}$ continuous.^[12]
• ${\displaystyle \tau _{\mathcal {L}}}$ is Hausdorff if and only if for every non-zero ${\displaystyle x\in X,}$ there exists some ${\displaystyle p\in {\mathcal {L}}}$ such that ${\displaystyle p(x)>0.}$^[12]
• If ${\displaystyle {\mathcal {F}}}$ is the set of all continuous F-seminorms on ${\displaystyle \left(X,\tau _{\mathcal {L}}\right)}$ then ${\displaystyle \tau _{\mathcal {L}}=\tau _{\mathcal {F}}.}$^[12]
• If ${\displaystyle {\mathcal {F}}}$ is the set of all pointwise suprema of non-empty finite subsets of ${\displaystyle {\mathcal {F}}}$ of ${\displaystyle {\mathcal {L}}}$ then ${\displaystyle {\mathcal {F}}}$ is a directed family of F-seminorms and ${\displaystyle \tau _{\mathcal {L}}=\tau _{\mathcal {F}}.}$^[12]

Fréchet combination

Suppose that ${\displaystyle p_{\bullet }=\left(p_{i}\right)_{i=1}^{\infty }}$ is a family of non-negative subadditive functions on a vector space ${\displaystyle X.}$

The Fréchet combination^[8] of ${\displaystyle p_{\bullet }}$ is defined to be the real-valued map

As an F-seminorm

Assume that ${\displaystyle p_{\bullet }=\left(p_{i}\right)_{i=1}^{\infty }}$ is an increasing sequence of seminorms on ${\displaystyle X}$ and let ${\displaystyle p}$ be the Fréchet combination of ${\displaystyle p_{\bullet }.}$ Then ${\displaystyle p}$ is an F-seminorm on ${\displaystyle X}$ that induces the same locally convex topology as the family ${\displaystyle p_{\bullet }}$ of seminorms.^[13]

Since ${\displaystyle p_{\bullet }=\left(p_{i}\right)_{i=1}^{\infty }}$ is increasing, a basis of open neighborhoods of the origin consists of all sets of the form ${\displaystyle \left\{x\in X~:~p_{i}(x)<r\right\}}$ as ${\displaystyle i}$ ranges over all positive integers and ${\displaystyle r>0}$ ranges over all positive real numbers.

The translation invariant pseudometric on ${\displaystyle X}$ induced by this F-seminorm ${\displaystyle p}$ 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 ${\displaystyle p_{i}}$ is a paranorm then so is ${\displaystyle p}$ and moreover, ${\displaystyle p}$ induces the same topology on ${\displaystyle X}$ as the family ${\displaystyle p_{\bullet }}$ of paranorms.^[8] This is also true of the following paranorms on ${\displaystyle X}$:

• ${\displaystyle q(x):=\inf _{}\left\{\sum _{i=1}^{n}p_{i}(x)+{\frac {1}{n}}~:~n>0{\text{ is an integer }}\right\}.}$^[8]
• ${\displaystyle r(x):=\sum _{n=1}^{\infty }\min \left\{{\frac {1}{2^{n}}},p_{n}(x)\right\}.}$^[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 ${\displaystyle R:[0,\infty )\to [0,\infty )}$ that has a bounded range, is subadditive (meaning that ${\displaystyle R(s+t)\leq R(s)+R(t)}$ for all ${\displaystyle s,t\geq 0}$), and satisfies ${\displaystyle R(s)=0}$ if and only if ${\displaystyle s=0.}$

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

Suppose that ${\displaystyle p_{\bullet }=\left(p_{i}\right)_{i=1}^{\infty }}$ is a family of non-negative F-seminorm on a vector space ${\displaystyle X,}$ ${\displaystyle R}$ is a bounded remetrization function, and ${\displaystyle r_{\bullet }=\left(r_{i}\right)_{i=1}^{\infty }}$ is a sequence of positive real numbers whose sum is finite. Then

defines a bounded F-seminorm that is uniformly equivalent to the ${\displaystyle p_{\bullet }.}$^[16] It has the property that for any net ${\displaystyle x_{\bullet }=\left(x_{a}\right)_{a\in A}}$ in ${\displaystyle X,}$ ${\displaystyle p\left(x_{\bullet }\right)\to 0}$ if and only if ${\displaystyle p_{i}\left(x_{\bullet }\right)\to 0}$ for all ${\displaystyle i.}$^[16] ${\displaystyle p}$ is an F-norm if and only if the ${\displaystyle p_{\bullet }}$ separate points on ${\displaystyle X.}$^[16]

Characterizations

Of (pseudo)metrics induced by (semi)norms

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

Of pseudometrizable TVS

If ${\displaystyle (X,\tau )}$ is a topological vector space (TVS) (where note in particular that ${\displaystyle \tau }$ is assumed to be a vector topology) then the following are equivalent:^[11]

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

Of metrizable TVS

If ${\displaystyle (X,\tau )}$ is a TVS then the following are equivalent:

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

Of locally convex pseudometrizable TVS

If ${\displaystyle (X,\tau )}$ is TVS then the following are equivalent:^[13]

1. ${\displaystyle X}$ is locally convex and pseudometrizable.
2. ${\displaystyle X}$ has a countable neighborhood base at the origin consisting of convex sets.
3. The topology of ${\displaystyle X}$ is induced by a countable family of (continuous) seminorms.
4. The topology of ${\displaystyle X}$ is induced by a countable increasing sequence of (continuous) seminorms ${\displaystyle \left(p_{i}\right)_{i=1}^{\infty }}$ (increasing means that for all ${\displaystyle i,}$ ${\displaystyle p_{i}\geq p_{i+1}.}$
5. The topology of ${\displaystyle X}$ is induced by an F-seminorm of the form:
   $${\displaystyle p(x)=\sum _{n=1}^{\infty }2^{-n}\operatorname {arctan} p_{n}(x)}$$

   
   where ${\displaystyle \left(p_{i}\right)_{i=1}^{\infty }}$ are (continuous) seminorms on ${\displaystyle X.}$^[18]

Quotients

Let ${\displaystyle M}$ be a vector subspace of a topological vector space ${\displaystyle (X,\tau ).}$

• If ${\displaystyle X}$ is a pseudometrizable TVS then so is ${\displaystyle X/M.}$^[11]
• If ${\displaystyle X}$ is a complete pseudometrizable TVS and ${\displaystyle M}$ is a closed vector subspace of ${\displaystyle X}$ then ${\displaystyle X/M}$ is complete.^[11]
• If ${\displaystyle X}$ is metrizable TVS and ${\displaystyle M}$ is a closed vector subspace of ${\displaystyle X}$ then ${\displaystyle X/M}$ is metrizable.^[11]
• If ${\displaystyle p}$ is an F-seminorm on ${\displaystyle X,}$ then the map ${\displaystyle P:X/M\to \mathbb {R} }$ defined by
  $${\displaystyle P(x+M):=\inf _{}\{p(x+m):m\in M\}}$$

  
  is an F-seminorm on ${\displaystyle X/M}$ that induces the usual quotient topology on ${\displaystyle X/M.}$^[11] If in addition ${\displaystyle p}$ is an F-norm on ${\displaystyle X}$ and if ${\displaystyle M}$ is a closed vector subspace of ${\displaystyle X}$ then ${\displaystyle P}$ is an F-norm on ${\displaystyle X.}$^[11]

Examples and sufficient conditions

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

If ${\displaystyle X}$ is Hausdorff locally convex TVS then ${\displaystyle X}$ with the strong topology, ${\displaystyle \left(X,b\left(X,X^{\prime }\right)\right),}$ is metrizable if and only if there exists a countable set ${\displaystyle {\mathcal {B}}}$ of bounded subsets of ${\displaystyle X}$ such that every bounded subset of ${\displaystyle X}$ is contained in some element of ${\displaystyle {\mathcal {B}}.}$^[22]

The strong dual space ${\displaystyle X_{b}^{\prime }}$ of a metrizable locally convex space (such as a Fréchet space^[23]) ${\displaystyle X}$ is a DF-space.^[24] The strong dual of a DF-space is a Fréchet space.^[25] The strong dual of a reflexive Fréchet space is a bornological space.^[24] The strong bidual (that is, the strong dual space of the strong dual space) of a metrizable locally convex space is a Fréchet space.^[26] If ${\displaystyle X}$ is a metrizable locally convex space then its strong dual ${\displaystyle X_{b}^{\prime }}$ has one of the following properties, if and only if it has all of these properties: (1) bornological, (2) infrabarreled, (3) barreled.^[26]

Normability

A topological vector space is seminormable if and only if it has a convex bounded neighborhood of the origin. Moreover, a TVS is normable if and only if it is Hausdorff and seminormable.^[14] Every metrizable TVS on a finite-dimensional vector space is a normable locally convex complete TVS, being TVS-isomorphic to Euclidean space. Consequently, any metrizable TVS that is not normable must be infinite dimensional.

If ${\displaystyle M}$ is a metrizable locally convex TVS that possess a countable fundamental system of bounded sets, then ${\displaystyle M}$ is normable.^[27]

If ${\displaystyle X}$ is a Hausdorff locally convex space then the following are equivalent:

1. ${\displaystyle X}$ is normable.
2. ${\displaystyle X}$ has a (von Neumann) bounded neighborhood of the origin.
3. the strong dual space ${\displaystyle X_{b}^{\prime }}$ of ${\displaystyle X}$ is normable.^[28]

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

4. the strong dual space of ${\displaystyle X}$ is metrizable.^[28]
5. the strong dual space of ${\displaystyle X}$ is a Fréchet–Urysohn locally convex space.^[23]

In particular, if a metrizable locally convex space ${\displaystyle X}$ (such as a Fréchet space) is not normable then its strong dual space ${\displaystyle X_{b}^{\prime }}$ is not a Fréchet–Urysohn space and consequently, this complete Hausdorff locally convex space ${\displaystyle X_{b}^{\prime }}$ is also neither metrizable nor normable.

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

Metrically bounded sets and bounded sets

Suppose that ${\displaystyle (X,d)}$ is a pseudometric space and ${\displaystyle B\subseteq X.}$ The set ${\displaystyle B}$ is metrically bounded or ${\displaystyle d}$-bounded if there exists a real number ${\displaystyle R>0}$ such that ${\displaystyle d(x,y)\leq R}$ for all ${\displaystyle x,y\in B}$; the smallest such ${\displaystyle R}$ is then called the diameter or ${\displaystyle d}$-diameter of ${\displaystyle B.}$^[14] If ${\displaystyle B}$ is bounded in a pseudometrizable TVS ${\displaystyle X}$ 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

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

Completeness

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

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

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

Subsets and subsequences

• Let ${\displaystyle M}$ be a separable locally convex metrizable topological vector space and let ${\displaystyle C}$ be its completion. If ${\displaystyle S}$ is a bounded subset of ${\displaystyle C}$ then there exists a bounded subset ${\displaystyle R}$ of ${\displaystyle X}$ such that ${\displaystyle S\subseteq \operatorname {cl} _{C}R.}$^[41]
• Every totally bounded subset of a locally convex metrizable TVS ${\displaystyle X}$ is contained in the closed convex balanced hull of some sequence in ${\displaystyle X}$ that converges to ${\displaystyle 0.}$
• In a pseudometrizable TVS, every bornivore is a neighborhood of the origin.^[42]
• If ${\displaystyle d}$ is a translation invariant metric on a vector space ${\displaystyle X,}$ then ${\displaystyle d(nx,0)\leq nd(x,0)}$ for all ${\displaystyle x\in X}$ and every positive integer ${\displaystyle n.}$^[43]
• If ${\displaystyle \left(x_{i}\right)_{i=1}^{\infty }}$ is a null sequence (that is, it converges to the origin) in a metrizable TVS then there exists a sequence ${\displaystyle \left(r_{i}\right)_{i=1}^{\infty }}$ of positive real numbers diverging to ${\displaystyle \infty }$ such that ${\displaystyle \left(r_{i}x_{i}\right)_{i=1}^{\infty }\to 0.}$^[43]
• A subset of a complete metric space is closed if and only if it is complete. If a space ${\displaystyle X}$ is not complete, then ${\displaystyle X}$ is a closed subset of ${\displaystyle X}$ that is not complete.
• If ${\displaystyle X}$ is a metrizable locally convex TVS then for every bounded subset ${\displaystyle B}$ of ${\displaystyle X,}$ there exists a bounded disk ${\displaystyle D}$ in ${\displaystyle X}$ such that ${\displaystyle B\subseteq X_{D},}$ and both ${\displaystyle X}$ and the auxiliary normed space ${\displaystyle X_{D}}$ induce the same subspace topology on ${\displaystyle B.}$^[44]

Generalized series

As described in this article's section on generalized series, for any ${\displaystyle I}$-indexed family family ${\displaystyle \left(r_{i}\right)_{i\in I}}$ of vectors from a TVS ${\displaystyle X,}$ it is possible to define their sum ${\displaystyle \textstyle \sum \limits _{i\in I}r_{i}}$ as the limit of the net of finite partial sums ${\displaystyle F\in \operatorname {FiniteSubsets} (I)\mapsto \textstyle \sum \limits _{i\in F}r_{i}}$ where the domain ${\displaystyle \operatorname {FiniteSubsets} (I)}$ is directed by ${\displaystyle \,\subseteq .\,}$ If ${\displaystyle I=\mathbb {N} }$ and ${\displaystyle X=\mathbb {R} ,}$ for instance, then the generalized series ${\displaystyle \textstyle \sum \limits _{i\in \mathbb {N} }r_{i}}$ converges if and only if ${\displaystyle \textstyle \sum \limits _{i=1}^{\infty }r_{i}}$ converges unconditionally in the usual sense (which for real numbers, is equivalent to absolute convergence). If a generalized series ${\displaystyle \textstyle \sum \limits _{i\in I}r_{i}}$ converges in a metrizable TVS, then the set ${\displaystyle \left\{i\in I:r_{i}\neq 0\right\}}$ is necessarily countable (that is, either finite or countably infinite);^{[proof 1]} in other words, all but at most countably many ${\displaystyle r_{i}}$ will be zero and so this generalized series ${\displaystyle \textstyle \sum \limits _{i\in I}r_{i}~=~\textstyle \sum \limits _{\stackrel {i\in I}{r_{i}\neq 0}}r_{i}}$ is actually a sum of at most countably many non-zero terms.

Linear maps

If ${\displaystyle X}$ is a pseudometrizable TVS and ${\displaystyle A}$ maps bounded subsets of ${\displaystyle X}$ to bounded subsets of ${\displaystyle Y,}$ then ${\displaystyle A}$ is continuous.^[14] Discontinuous linear functionals exist on any infinite-dimensional pseudometrizable TVS.^[46] Thus, a pseudometrizable TVS is finite-dimensional if and only if its continuous dual space is equal to its algebraic dual space.^[46]

If ${\displaystyle F:X\to Y}$ is a linear map between TVSs and ${\displaystyle X}$ is metrizable then the following are equivalent:

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

Open and almost open maps

Theorem: If ${\displaystyle X}$ is a complete pseudometrizable TVS, ${\displaystyle Y}$ is a Hausdorff TVS, and ${\displaystyle T:X\to Y}$ is a closed and almost open linear surjection, then ${\displaystyle T}$ is an open map.^[47]

Theorem: If ${\displaystyle T:X\to Y}$ is a surjective linear operator from a locally convex space ${\displaystyle X}$ onto a barrelled space ${\displaystyle Y}$ (e.g. every complete pseudometrizable space is barrelled) then ${\displaystyle T}$ is almost open.^[47]

Theorem: If ${\displaystyle T:X\to Y}$ is a surjective linear operator from a TVS ${\displaystyle X}$ onto a Baire space ${\displaystyle Y}$ then ${\displaystyle T}$ is almost open.^[47]

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

Hahn-Banach extension property

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

The Hahn-Banach theorem guarantees that every Hausdorff locally convex space has the HBEP. For complete metrizable TVSs there is a converse:

If a vector space ${\displaystyle X}$ has uncountable dimension and if we endow it with the finest vector topology then this is a TVS with the HBEP that is neither locally convex or metrizable.^[22]

See also

• Asymmetric norm – Generalization of the concept of a norm
• Complete metric space – Metric geometry
• Complete topological vector space – A TVS where points that get progressively closer to each other will always converge to a point
• Equivalence of metrics
• F-space – Topological vector space with a complete translation-invariant metric
• Fréchet space – A locally convex topological vector space that is also a complete metric space
• Generalised metric – Metric geometry
• K-space (functional analysis)
• Locally convex topological vector space – A vector space with a topology defined by convex open sets
• Metric space – Mathematical space with a notion of distance
• Pseudometric space – Generalization of metric spaces in mathematics
• Relation of norms and metrics – Mathematical space with a notion of distancePages displaying short descriptions of redirect targets
• Seminorm – nonnegative-real-valued function on a real or complex vector space that satisfies the triangle inequality and is absolutely homogenousPages displaying wikidata descriptions as a fallback
• Sublinear function – Type of function in linear algebra
• Uniform space – Topological space with a notion of uniform properties
• Ursescu theorem – Generalization of closed graph, open mapping, and uniform boundedness theorem

Notes

Proofs

References

Bibliography

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