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:

  • and for all and positive integers
  • The set is an additive subgroup of

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 [Lattice_(order)#Conditional_completeness|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
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

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 is a value on In particular, and for all

Topology induced by a family of ''F''-seminorms

Suppose that is a non-empty collection of F-seminorms on a vector space and for any finite subset and any let
The set forms a filter base on that also forms a neighborhood basis at the origin for a vector topology on denoted by Each is a balanced and absorbing subset of These sets satisfy

  • is the coarsest vector topology on making each continuous.
  • is Hausdorff if and only if for every non-zero there exists some such that
  • If is the set of all continuous F-seminorms on then
  • If is the set of all pointwise suprema of non-empty finite subsets of of then is a directed family of F-seminorms and

Fréchet combination

Suppose that is a family of non-negative subadditive functions on a vector space
The Fréchet combination of is defined to be the real-valued map

As an ''F''-seminorm

Assume that is an increasing sequence of seminorms on and let be the Fréchet combination of
Then is an F-seminorm on that induces the same locally convex topology as the family of seminorms.
Since is increasing, a basis of open neighborhoods of the origin consists of all sets of the form as ranges over all positive integers and ranges over all positive real numbers.
The translation invariant pseudometric on induced by this F-seminorm is
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 is a paranorm then so is and moreover, induces the same topology on as the family of paranorms.
This is also true of the following paranorms on :

Generalization

The Fréchet combination can be generalized by use of a bounded remetrization function.
A is a continuous non-negative non-decreasing map that has a bounded range, is subadditive, and satisfies if and only if
Examples of bounded remetrization functions include and
If is a pseudometric on and is a bounded remetrization function then is a bounded pseudometric on that is uniformly equivalent to
Suppose that is a family of non-negative F-seminorm on a vector space is a bounded remetrization function, and is a sequence of positive real numbers whose sum is finite.
Then
defines a bounded F-seminorm that is uniformly equivalent to the
It has the property that for any net in if and only if for all
is an F-norm if and only if the separate points on

Characterizations

Of (pseudo)metrics induced by (semi)norms

A pseudometric is induced by a seminorm on a vector space if and only if is translation invariant and absolutely homogeneous, which means that for all scalars and all in which case the function defined by is a seminorm and the pseudometric induced by is equal to

Of pseudometrizable TVS

If is a topological vector space then the following are equivalent:

  1. is pseudometrizable.
  2. has a countable neighborhood base at the origin.
  3. The topology on is induced by a translation-invariant pseudometric on
  4. The topology on is induced by an F-seminorm.
  5. The topology on is induced by a paranorm.

Of metrizable TVS

If is a TVS then the following are equivalent:

  1. is metrizable.
  2. is Hausdorff and pseudometrizable.
  3. is Hausdorff and has a countable neighborhood base at the origin.
  4. The topology on is induced by a translation-invariant metric on
  5. The topology on is induced by an F-norm.
  6. The topology on is induced by a monotone F-norm.
  7. The topology on is induced by a total paranorm.

Of locally convex pseudometrizable TVS

If is TVS then the following are equivalent:

  1. is locally convex and pseudometrizable.
  2. has a countable neighborhood base at the origin consisting of convex sets.
  3. The topology of is induced by a countable family of seminorms.
  4. The topology of is induced by a countable increasing sequence of seminorms seminorms on

Quotients

Let be a vector subspace of a topological vector space

  • If is a pseudometrizable TVS then so is
  • If is a complete pseudometrizable TVS and is a closed vector subspace of then is complete.
  • If is metrizable TVS and is a closed vector subspace of then is metrizable.
  • If is an F-seminorm on then the map defined by
    is an F-seminorm on that induces the usual quotient topology on If in addition is an F-norm on and if is a closed vector subspace of then is an F-norm on

Examples and sufficient conditions


  • Every seminormed space is pseudometrizable with a canonical pseudometric given by for all.
  • If is pseudometric TVS with a translation invariant pseudometric then defines a paranorm. However, if is a translation invariant pseudometric on the vector space, then 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 is either a DF-space or an LM-space. If is a sequential space then it is either metrizable or else a Montel DF-space.
If is Hausdorff locally convex TVS then with the strong topology, is metrizable if and only if there exists a countable set of bounded subsets of such that every bounded subset of is contained in some element of
The strong dual space of a metrizable locally convex space 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 of a metrizable locally convex space is a Fréchet space.
If is a metrizable locally convex space then its strong dual has one of the following properties, if and only if it has all of these properties: bornological, infrabarreled, 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 normable must be infinite dimensional.
If is a metrizable locally convex TVS that possess a countable fundamental system of bounded sets, then is normable.
If is a Hausdorff locally convex space then the following are equivalent:

  1. is normable.
  2. has a (von Neumann) bounded neighborhood of the origin.
  3. the strong dual space of is normable.
and if this locally convex space is also metrizable, then the following may be appended to this list:

  1. the strong dual space of is metrizable.
  2. the strong dual space of is a Fréchet–Urysohn locally convex space.
In particular, if a metrizable locally convex space is normable then its strong dual space is not a Fréchet–Urysohn space and consequently, this complete Hausdorff locally convex space is also neither metrizable nor normable.
Another consequence of this is that if is a reflexive locally convex TVS whose strong dual is metrizable then is necessarily a reflexive Fréchet space, is a DF-space, both and are necessarily complete Hausdorff ultrabornological distinguished webbed spaces, and moreover, is normable if and only if is normable if and only if is Fréchet–Urysohn if and only if is metrizable. In particular, such a space is either a Banach space or else it is not even a Fréchet–Urysohn space.

Metrically bounded sets and bounded sets

Suppose that is a pseudometric space and
The set is metrically bounded or -bounded if there exists a real number such that for all ;
the smallest such is then called the diameter or -diameter of
If is bounded in a pseudometrizable TVS then it is metrically bounded;
the converse is in general false but it is true for locally convex metrizable TVSs.

Properties of pseudometrizable TVS


  • Every metrizable locally convex TVS is a quasibarrelled space, bornological space, and a Mackey space.
  • Every complete metrizable TVS is a barrelled space and a Baire space. However, there exist metrizable Baire spaces that are not complete.
  • If is a metrizable locally convex space, then the strong dual of is bornological if and only if it is barreled, if and only if it is infrabarreled.
  • If is a complete pseudometrizable TVS and is a closed vector subspace of then is complete.
  • The strong dual of a locally convex metrizable TVS is a webbed space.
  • If and are complete metrizable TVSs and if is coarser than then ; this is no longer guaranteed to be true if any one of these metrizable TVSs is not complete. Said differently, if and are both F-spaces but with different topologies, then neither one of and contains the other as a subset. One particular consequence of this is, for example, that if is a Banach space and is some other normed space whose norm-induced topology is finer than that of, then the only way that can be a Banach space is if these two norms and are equivalent; if they are not equivalent, then can not be a Banach space.
    As another consequence, if is a Banach space and is a Fréchet space, then the map is continuous if and only if the Fréchet space the TVS .
  • 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
  • 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 metrizable TVS is a barrelled space and a Baire space.
  • The dimension of a complete metrizable TVS is either finite or uncountable.

Completeness

Every topological vector space has a canonical uniform structure, induced by its topology, which allows the notions of completeness and uniform continuity to be applied to it.
If is a metrizable TVS and is a metric that defines 's topology, then its possible that is complete as a TVS but the metric is a complete metric.
Thus, if is a TVS whose topology is induced by a pseudometric then the notion of completeness of and the notion of completeness of the pseudometric space are not always equivalent.
The next theorem gives a condition for when they are equivalent:
If is a closed vector subspace of a complete pseudometrizable TVS then the quotient space is complete.
If is a vector subspace of a metrizable TVS and if the quotient space is complete then so is If is not complete then but not complete, vector subspace of
A Baire separable topological group is metrizable if and only if it is cosmic.

Subsets and subsequences


  • Let be a separable locally convex metrizable topological vector space and let be its completion. If is a bounded subset of then there exists a bounded subset of such that
  • Every totally bounded subset of a locally convex metrizable TVS is contained in the closed convex balanced hull of some sequence in that converges to
  • In a pseudometrizable TVS, every bornivore is a neighborhood of the origin.
  • If is a translation invariant metric on a vector space then for all and every positive integer
  • If is a null sequence in a metrizable TVS then there exists a sequence of positive real numbers diverging to such that
  • A subset of a complete metric space is closed if and only if it is complete. If a space is not complete, then is a closed subset of that is not complete.
  • If is a metrizable locally convex TVS then for every bounded subset of there exists a bounded disk in such that and both and the auxiliary normed space induce the same subspace topology on
Generalized series
As described in this article's section on generalized series, for any -indexed family family of vectors from a TVS it is possible to define their sum as the limit of the net of finite partial sums where the domain is directed by
If and for instance, then the generalized series converges if and only if converges unconditionally in the usual sense.
If a generalized series converges in a metrizable TVS, then the set is necessarily countable ;
in other words, all but at most countably many will be zero and so this generalized series is actually a sum of at most countably many non-zero terms.

Linear maps

If is a pseudometrizable TVS and maps bounded subsets of to bounded subsets of then is continuous.
Discontinuous linear functionals exist on any infinite-dimensional pseudometrizable TVS. Thus, a pseudometrizable TVS is finite-dimensional if and only if its continuous dual space is equal to its algebraic dual space.
If is a linear map between TVSs and is metrizable then the following are equivalent:

  1. is continuous;
  2. is a bounded map ;
  3. is sequentially continuous;
  4. the image under of every null sequence in is a bounded set where by definition, a is a sequence that converges to the origin.
  5. maps null sequences to null sequences;
'''Open and almost open maps'''

Hahn-Banach extension property

A vector subspace of a TVS has the extension property if any continuous linear functional on can be extended to a continuous linear functional on
Say that a TVS has the Hahn-Banach extension property if every vector subspace of has the extension property.
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 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.