Metrizable topological vector space

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 ${\displaystyle d.}$[1]

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. ${\displaystyle d(x,y)=d(x-y,0){\text{ for all }}x,y\in X.}$

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 ${\displaystyle x=0.}$

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]

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.

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.

• 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]

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]

• 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]

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 ${\displaystyle x\in X.}$

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 ${\displaystyle \tau _{\mathcal {L}}.}$[12] Each ${\displaystyle U_{{\mathcal {F}},r}}$ is a balanced and absorbing subset of ${\displaystyle X.}$[12] These sets satisfy[12]

$${\displaystyle U_{{\mathcal {F}},r/2}+U_{{\mathcal {F}},r/2}\subseteq U_{{\mathcal {F}},r}.}$$

• ${\displaystyle \tau _{\mathcal {L}}}$ is the coarsest vector topology on ${\displaystyle X}$ making each ${\displaystyle p\in {\mathcal {L}}}$ continuous.[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]

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

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

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

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]

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

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

defines a bounded F-seminorm that is uniformly equivalent to the ${\displaystyle p_{\bullet }.}$[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]

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 ${\displaystyle d.}$

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.

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.