Metrizable topological vector space


In functional analysis and related areas of mathematics, a metrizable topological vector spaces is a TVS whose topology is induced by a metric.

Pseudometrics and metrics

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

  1. for all ;
  2. Symmetry: for all ;
  3. Subadditivity: for all.
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: for all,.
;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:
forms a basis for a topology on that is called the pseudometric topology or -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

Every topological vector space is an additive commutative topological group but not all group topologies on are vector topologies.
This is because a group topology on a vector space may 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: for all ;
  2. for all.

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: for all ;
  3. .
  4. Symmetric: for all.
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 for all such that.
The topology that induces on is the discrete topology, which makes into a commutative topological group under addition but does not 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 not enough to guarantee a vector topology, which leads us to define paranorms and -seminorms.

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 for all.
  4. Separate continuity:
    • if for some scalar then for every ;
    • if is a scalar,, and then.
A paranorm is called an total if in addition it satisfies:

Properties of paranorms

If is a paranorm on a vector space then:

Examples of paranorms

''F''-seminorms

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

  1. Non-negative:.
  2. Subadditive: for all ;
  3. Balanced: for all and all scalars satisfying ;
    • This condition guarantees that each set of the form or for some is balanced.
  4. for every, as
    • The sequence can be replaced by any positive sequence converging to 0.
An -seminorm is called an -norm if in addition it satisfies:

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

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

''F''-seminormed spaces

Every isometric embedding of one -seminormed space into another is a topological embedding, but the converse is not true in general.

Examples of ''F''-seminorms

Properties of ''F''-seminorms

Topology induced by a single ''F''-seminorm

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

Suppose that is a non-empty collection of -seminorms on a vector space and for any finite subset and any, let
The set forms a filter base on and this filter base forms a neighborhood basis at the origin for a vector topology on.

Fréchet combination

Suppose that is a family of non-negative subadditive functions on a vector space.
Definition: 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.
Then is an -seminorm on that induces the same locally convex topology as the family of seminorms.
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 -seminorm is the
.

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 :

Characterizations

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 -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 -norm.
  6. The topology on is induced by a monotone -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 .
  5. The topology of is induced by an -seminorm of the form:
    where are seminorms on.

Quotients

Let be a vector subspace of a topological vector space.

Examples and sufficient conditions

If is a Hausdorff locally convex TVS then the following are equivalent:

  1. is normable.
  2. has a bounded neighborhood of the origin.
  3. the strong dual of is normable.
  4. the strong dual of is metrizable.
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.

Metrically bounded sets and bounded sets

Suppose that is a pseudometric space and.
We say that 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 bornological space and a Mackey space.
If is a metrizable locally convex space, then the strong dual of is bornological if and only if it is infrabarreled, if and only if it is barreled.
If is a complete pseudo-metrizable 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 true if either one of these metrizable TVSs is not complete.

Completeness

Recall that 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 not 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:

Subsets and subsequences

Linear maps

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;
    • Recall that a null sequence is a sequence that converges to the origin.
  5. maps null sequences to null sequences;

Hahn-Banach extension property

Let be a TVS.
Say that a vector subspace of has the extension property if any continuous linear functional on can be extended to a continuous linear functional on.
Say that 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.