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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: $${\displaystyle B_{r}(z):=\{x\in X:d(x,z)<r\}}$$ 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 ${\displaystyle d.}$

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:

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:

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

Equivalence on topological groups

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

Pseudometrizable topological groups

Theorem — If ${\displaystyle (X,\tau )}$ is an additive commutative topological group then the following are equivalent:

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

If ${\displaystyle (X,\tau )}$ 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 ${\displaystyle X}$ 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 if for every ${\displaystyle N\in {\mathcal {N}},}$ there exists some ${\displaystyle U\in {\mathcal {N}}}$ such that ${\displaystyle U+U\subseteq N.}$

Continuity of addition at 0 — If ${\displaystyle (X,+)}$ is a group (as all vector spaces are), ${\displaystyle \tau }$ is a topology on ${\displaystyle X,}$ and ${\displaystyle X\times X}$ is endowed with the product topology, then the addition map ${\displaystyle X\times X\to X}$ (i.e. the map ${\displaystyle (x,y)\mapsto x+y}$) is continuous at the origin of ${\displaystyle X\times X}$ if and only if the set of neighborhoods of the origin in ${\displaystyle (X,\tau )}$ 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 non-negative continuous real-valued subadditive functions. These functions can then be used to prove many of the basic properties of topological vector spaces and also show that a Hausdorff TVS with a countable basis of neighborhoods is metrizable. The following theorem is true more generally for commutative additive topological groups.

Theorem — Let ${\displaystyle U_{\bullet }=\left(U_{i}\right)_{i=0}^{\infty }}$ be a collection of subsets of a vector space such that ${\displaystyle 0\in U_{i}}$ and ${\displaystyle U_{i+1}+U_{i+1}\subseteq U_{i}}$ for all ${\displaystyle i\geq 0.}$ For all ${\displaystyle u\in U_{0},}$ let $${\displaystyle \mathbb {S} (u):=\left\{n_{\bullet }=\left(n_{1},\ldots ,n_{k}\right)~:~k\geq 1,n_{i}\geq 0{\text{ for all }}i,{\text{ and }}u\in U_{n_{1}}+\cdots +U_{n_{k}}\right\}.}$$

Define ${\displaystyle f:X\to }$ by ${\displaystyle f(x)=1}$ if ${\displaystyle x\not \in U_{0}}$ and otherwise let $${\displaystyle f(x):=\inf _{}\left\{2^{-n_{1}}+\cdots 2^{-n_{k}}~:~n_{\bullet }=\left(n_{1},\ldots ,n_{k}\right)\in \mathbb {S} (x)\right\}.}$$

Then ${\displaystyle f}$ is subadditive (meaning ${\displaystyle f(x+y)\leq f(x)+f(y){\text{ for all }}x,y\in X}$) and ${\displaystyle f=0}$ on ${\displaystyle \bigcap _{i\geq 0}U_{i},}$ so in particular ${\displaystyle f(0)=0.}$ If all ${\displaystyle U_{i}}$ are symmetric sets then ${\displaystyle f(-x)=f(x)}$ and if all ${\displaystyle U_{i}}$ are balanced then ${\displaystyle f(sx)\leq f(x)}$ for all scalars ${\displaystyle s}$ such that ${\displaystyle |s|\leq 1}$ and all ${\displaystyle x\in X.}$ If ${\displaystyle X}$ is a topological vector space and if all ${\displaystyle U_{i}}$ are neighborhoods of the origin then ${\displaystyle f}$ is continuous, where if in addition ${\displaystyle X}$ is Hausdorff and ${\displaystyle U_{\bullet }}$ forms a basis of balanced neighborhoods of the origin in ${\displaystyle X}$ then ${\displaystyle d(x,y):=f(x-y)}$ is a metric defining the vector topology on ${\displaystyle X.}$

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 ${\displaystyle s_{\bullet }=\left(s_{i}\right)_{i=1}^{\infty }}$":

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:
   • 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 ${\displaystyle X.}$

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 ${\displaystyle X.}$
• ${\displaystyle p(x+n)=p(x){\text{ for all }}x,n\in X}$ with ${\displaystyle p(n)=0.}$
• 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) 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 ${\displaystyle \tau .}$
• If ${\displaystyle p}$ is a paranorm on ${\displaystyle X}$ then so is the map ${\displaystyle q(x):=p(x)/.}$
• Every positive scalar multiple of a paranorm (resp. total paranorm) is again such a paranorm (resp. total paranorm).
• Every seminorm is a paranorm.
• The restriction of an paranorm (resp. total paranorm) to a vector subspace is an paranorm (resp. total paranorm).
• The sum of two paranorms is a paranorm.
• 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.
• 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 paranorms on ${\displaystyle X:=\mathbb {R} ^{2}.}$
• 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 ${\displaystyle s.}$
• 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.
• 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 ${\displaystyle X_{\mathbb {R} }.}$

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:

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.

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 ${\displaystyle s<t.}$

F-seminormed spaces

An F-seminormed space (resp. F-normed space) 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 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.

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 ${\displaystyle X.}$
• The restriction of an F-seminorm (resp. F-norm) to a vector subspace is an F-seminorm (resp. F-norm).
• 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. 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 $${\displaystyle ^{p}=\left|x_{1}\right|^{p}+\cdots \left|x_{n}\right|^{p}}$$ 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 ${\displaystyle X.}$
• 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 ${\displaystyle X_{\mathbb {R} }.}$

Properties of F-seminorms

Every F-seminorm is a paranorm and every paranorm is equivalent to some F-seminorm. 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

Theorem — Let ${\displaystyle p}$ be an F-seminorm on a vector space ${\displaystyle X.}$ Then the map ${\displaystyle d:X\times X\to \mathbb {R} }$ defined by ${\displaystyle d(x,y):=p(x-y)}$ is a translation invariant pseudometric on ${\displaystyle X}$ that defines a vector topology ${\displaystyle \tau }$ on ${\displaystyle X.}$ If ${\displaystyle p}$ is an F-norm then ${\displaystyle d}$ is a metric. When ${\displaystyle X}$ is endowed with this topology then ${\displaystyle p}$ is a continuous map on ${\displaystyle X.}$

The balanced sets ${\displaystyle \{x\in X~:~p(x)\leq r\},}$ as ${\displaystyle r}$ ranges over the positive reals, form a neighborhood basis at the origin for this topology consisting of closed set. Similarly, the balanced sets ${\displaystyle \{x\in X~:~p(x)<r\},}$ as ${\displaystyle r}$ 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 ${\displaystyle {\mathcal {L}}}$ 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 $${\displaystyle U_{{\mathcal {F}},r}:=\bigcap _{p\in {\mathcal {F}}}\{x\in X:p(x)<r\}.}$$

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 ${\displaystyle \tau _{\mathcal {L}}.}$ Each ${\displaystyle U_{{\mathcal {F}},r}}$ is a balanced and absorbing subset of ${\displaystyle X.}$ These sets satisfy $${\displaystyle U_{{\mathcal {F}},r/2}+U_{{\mathcal {F}},r/2}\subseteq U_{{\mathcal {F}},r}.}$$

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

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 of ${\displaystyle p_{\bullet }}$ is defined to be the real-valued map $${\displaystyle p(x):=\sum _{i=1}^{\infty }{\frac {p_{i}(x)}{2^{i}\left}}.}$$

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.

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

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

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. This is also true of the following paranorms on ${\displaystyle X}$:

• ${\displaystyle q(x):=\inf _{}\left\{\sum _{i=1}^{n}p_{i}(x)+{\frac {1}{n}}~:~n>0{\text{ is an integer }}\right\}.}$
• ${\displaystyle r(x):=\sum _{n=1}^{\infty }\min \left\{{\frac {1}{2^{n}}},p_{n}(x)\right\}.}$

Generalization

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

A bounded remetrization function 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 ${\displaystyle t\mapsto {\frac {t}{1+t}}.}$ 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 ${\displaystyle d.}$

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 $${\displaystyle p(x):=\sum _{i=1}^{\infty }r_{i}R\left(p_{i}(x)\right)}$$ defines a bounded F-seminorm that is uniformly equivalent to the ${\displaystyle p_{\bullet }.}$ 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 ${\displaystyle i.}$ ${\displaystyle p}$ is an F-norm if and only if the ${\displaystyle p_{\bullet }}$ separate points on ${\displaystyle X.}$

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:

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.
4. The topology on ${\displaystyle X}$ is induced by a translation-invariant metric on ${\displaystyle X.}$
5. The topology on ${\displaystyle X}$ is induced by an F-norm.
6. The topology on ${\displaystyle X}$ is induced by a monotone F-norm.
7. The topology on ${\displaystyle X}$ is induced by a total paranorm.

Birkhoff–Kakutani theorem — If ${\displaystyle (X,\tau )}$ is a topological vector space then the following three conditions are equivalent:

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

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

Of locally convex pseudometrizable TVS

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

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 ${\displaystyle X.}$

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 ${\displaystyle X/M.}$
• 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.
• If ${\displaystyle X}$ is metrizable TVS and ${\displaystyle M}$ is a closed vector subspace of ${\displaystyle X}$ then ${\displaystyle X/M}$ is metrizable.
• 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 ${\displaystyle X/M.}$ 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 ${\displaystyle X.}$

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 ${\displaystyle x,y\in X.}$.
• If ${\displaystyle (X,d)}$ is pseudometric TVS with a translation invariant pseudometric ${\displaystyle d,}$ then ${\displaystyle p(x):=d(x,0)}$ defines a paranorm. 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 nor a paranorm.
• If a TVS has a bounded neighborhood of the origin then it is pseudometrizable; the converse is in general false.
• If a Hausdorff TVS has a bounded neighborhood of the origin then it is metrizable.
• 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 ${\displaystyle {\mathcal {B}}.}$

The strong dual space ${\displaystyle X_{b}^{\prime }}$ of a metrizable locally convex space (such as a Fréchet space) ${\displaystyle X}$ is a DF-space. The strong dual of a DF-space is a Fréchet space. The strong dual of a reflexive Fréchet space is a bornological space. The strong bidual (that is, the strong dual space of the strong dual space) of a metrizable locally convex space is a Fréchet space. 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.

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. 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.

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.

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.
5. the strong dual space of ${\displaystyle X}$ is a Fréchet–Urysohn locally convex space.

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 ${\displaystyle B.}$ 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.

Properties of pseudometrizable TVS

Theorem — All infinite-dimensional separable complete metrizable TVS are homeomorphic.

• Every metrizable locally convex TVS is a quasibarrelled space, bornological space, and a Mackey space.
• Every complete pseudometrizable TVS is a barrelled space and a Baire space (and hence non-meager). However, there exist metrizable Baire spaces that are not complete.
• 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.
• 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.
• The strong dual of a locally convex metrizable TVS is a webbed space.
• 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 ${\displaystyle \tau =\nu }$; this is no longer guaranteed to be true if any one of these metrizable TVSs is not complete. 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.
• Any product of complete metrizable TVSs is a Baire space.
• A product of metrizable TVSs is metrizable if and only if it all but at most countably many of these TVSs have dimension ${\displaystyle 0.}$
• 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).
• The dimension of a complete metrizable TVS is either finite or uncountable.