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 edit

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

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]

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 edit

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 edit

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 edit

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 edit

Pseudometrizable topological groups edit

An invariant pseudometric that doesn't induce a vector topology edit

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.

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.

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 edit

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 edit

• 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]

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 edit

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 edit

• 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 edit

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 edit

Topology induced by a family of F-seminorms edit

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]

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 edit

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 edit

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 edit

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

Of (pseudo)metrics induced by (semi)norms edit

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 edit

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.