Metrizable topological vector space


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

Pseudometrics and metrics

A pseudometric on a set is a map satisfying the following properties:

  1. ;
  2. Symmetry: ;
  3. Subadditivity:
A pseudometric is called a metric if it satisfies:

  1. Identity of indiscernibles: for all if then
Ultrapseudometric
A pseudometric on is called a ultrapseudometric or a strong pseudometric if it satisfies:

  1. Strong/'Ultrametric triangle inequality:
Pseudometric space
A pseudometric space is a pair consisting of a set and a pseudometric on such that 's topology is identical to the topology on induced by We call a pseudometric space a metric space' when is a metric.

Topology induced by a pseudometric

If is a pseudometric on a set then collection of open balls:
as ranges over and ranges over the positive real numbers,
forms a basis for a topology on that is called the -topology or the pseudometric topology on induced by
Pseudometrizable space
A topological space is called pseudometrizable if there exists a pseudometric on such that is equal to the topology induced by

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 on a real or complex vector space is called a vector topology or a TVS topology if it makes the operations of vector addition and scalar multiplication continuous.
Every topological vector space is an additive commutative topological group but not all group topologies on are vector topologies.
This is because despite it making addition and negation continuous, a group topology on a vector space 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 is an additive group then we say that a pseudometric on is translation invariant or just invariant if it satisfies any of the following equivalent conditions:

  1. Translation invariance: ;

Value/G-seminorm

If is a topological group the a value or G-seminorm on is a real-valued map with the following properties:

  1. Non-negative:
  2. Subadditive: ;
  4. Symmetric:
where we call a G-seminorm a G-norm if it satisfies the additional condition:

  1. Total/'Positive definite': If then

Properties of values

If is a value on a vector space then:

Equivalence on topological groups

Pseudometrizable topological groups

An invariant pseudometric that doesn't induce a vector topology

Let be a non-trivial real or complex vector space and let be the translation-invariant trivial metric on defined by and such that
The topology that induces on is the discrete topology, which makes into a commutative topological group under addition but does form a vector topology on because is disconnected but every vector topology is connected.
What fails is that scalar multiplication isn't continuous on
This example shows that a translation-invariant metric is enough to guarantee a vector topology, which leads us to define paranorms and F-seminorms.

Additive sequences

A collection of subsets of a vector space is called additive if for every there exists some such that
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.
Assume that always denotes a finite sequence of non-negative integers and use the notation:
For any integers and
From this it follows that if consists of distinct positive integers then
It will now be shown by induction on that if consists of non-negative integers such that for some integer then
This is clearly true for and so assume that which implies that all are positive.
If all are distinct then this step is done, and otherwise pick distinct indices such that and construct from by replacing each with and deleting the element of .
Observe that and so by appealing to the inductive hypothesis we conclude that as desired.
It is clear that and that so to prove that is subadditive, it suffices to prove that when are such that which implies that
This is an exercise.
If all are symmetric then if and only if from which it follows that and
If all are balanced then the inequality for all unit scalars such that is proved similarly.
Because is a nonnegative subadditive function satisfying as described in the article on sublinear functionals, is uniformly continuous on if and only if is continuous at the origin.
If all are neighborhoods of the origin then for any real pick an integer such that so that implies
If the set of all form basis of balanced neighborhoods of the origin then it may be shown that for any there exists some such that implies

Paranorms

If is a vector space over the real or complex numbers then a paranorm on is a G-seminorm on that satisfies any of the following additional conditions, each of which begins with "for all sequences in and all convergent sequences of scalars ":

  1. Continuity of multiplication: if is a scalar and are such that and then
  2. Both of the conditions:
    • if and if is such that then ;
    • if then for every scalar
  3. Both of the conditions:
    • if and for some scalar then ;
    • if then
  4. Separate continuity:
    • if for some scalar then for every ;
    • if is a scalar, and then .
A paranorm is called total if in addition it satisfies:

  • Total/'Positive definite': implies

Properties of paranorms

If is a paranorm on a vector space then the map defined by is a translation-invariant pseudometric on that defines a on
If is a paranorm on a vector space then:

  • the set is a vector subspace of
  • with
  • If a paranorm satisfies and scalars then is absolutely homogeneity and thus is a seminorm.

Examples of paranorms


  • If is a translation-invariant pseudometric on a vector space that induces a vector topology on then the map defines a continuous paranorm on ; moreover, the topology that this paranorm defines in is
  • If is a paranorm on then so is the map
  • Every positive scalar multiple of a paranorm is again such a paranorm.
  • Every seminorm is a paranorm.
  • The restriction of an paranorm to a vector subspace is an paranorm.
  • The sum of two paranorms is a paranorm.
  • If and are paranorms on then so is Moreover, and This makes the set of paranorms on into a conditionally complete lattice.
  • Each of the following real-valued maps are paranorms on :
  • The real-valued maps and are paranorms on
  • If is a Hamel basis on a vector space then the real-valued map that sends to is a paranorm on which satisfies for all and scalars
  • The function is a paranorm on that is balanced but nevertheless equivalent to the usual norm on Note that the function is subadditive.
  • Let be a complex vector space and let denote considered as a vector space over Any paranorm on is also a paranorm on

''F''-seminorms

If is a vector space over the real or complex numbers then an F-seminorm on is a real-valued map with the following four properties:

  1. Non-negative:
  2. Subadditive: for all
  3. Balanced: for all scalars satisfying
    • This condition guarantees that each set of the form or for some is a balanced set.
  4. For every as
    • The sequence can be replaced by any positive sequence converging to the zero.
An F''-seminorm is called an F-norm if in addition it satisfies:

  1. Total/'Positive definite: implies
An F-seminorm is called monotone if it satisfies:

  1. Monotone''': for all non-zero and all real and such that

''F''-seminormed spaces

An F-seminormed space is a pair consisting of a vector space and an F-seminorm on
If and are F-seminormed spaces then a map is called an isometric embedding if
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 is again an F-seminorm.
  • The sum of finitely many F-seminorms is an F-seminorm.
  • If and are F-seminorms on then so is their pointwise supremum The same is true of the supremum of any non-empty finite family of F-seminorms on
  • The restriction of an F-seminorm to a vector subspace is an F-seminorm.
  • A non-negative real-valued function on 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 the map on defined by
    is an F-norm that is not a norm.
  • If is a linear map and if is an F-seminorm on then is an F-seminorm on
  • Let be a complex vector space and let denote considered as a vector space over Any F-seminorm on is also an F-seminorm on