# Metrizable locally convex space

In the mathematical discipline of functional analysis , topological vector spaces , i.e. vector spaces with a suitable topological structure, are examined. In particular, locally convex spaces are of interest, since Hahn-Banach's theorem guarantees a rich topological dual space for them. The important topological property of metrizability can be characterized in locally convex spaces by the zero neighborhood bases. Since the completion of a metrizable and locally convex space is a Fréchet space , such spaces are also called pre-F space .

## characterization

A locally convex space can be metrized if and only if it has a countable zero neighborhood basis . ${\ displaystyle E}$

If, in fact, there is a countable zero neighborhood base, it can be assumed to be absolutely convex without restriction . For each then defines a semi-norm on and ${\ displaystyle (U_ {n}) _ {n \ in {\ mathbb {N}}}}$${\ displaystyle U_ {n}}$${\ displaystyle n \ in {\ mathbb {N}}}$${\ displaystyle p_ {n} (x): = \ inf \ {t> 0; x \ in tU_ {n} \}}$${\ displaystyle E}$

${\ displaystyle d (x, y): = \ sum _ {n = 1} ^ {\ infty} {\ frac {1} {2 ^ {n}}} {\ frac {p_ {n} (xy)} {1 + p_ {n} (xy)}}}$

is a metric that generates the topology on . The converse is clear, since in a metric space each point has an enumerable neighborhood base. ${\ displaystyle E}$

## Examples

• Standardized spaces can be metrised and are locally convex.
• ${\ displaystyle {\ mathbb {R}} ^ {\ mathbb {N}}}$with the product topology is an example of a metrizable, locally convex space that cannot be normalized .
• Let be the vector space of the continuous functions . For be the semi-norm defined by . Then, with the topology defined by these semi-norms, it is even possible to metrize a Fréchet dream.${\ displaystyle C ({\ mathbb {R}})}$${\ displaystyle f \ colon {\ mathbb {R}} \ rightarrow {\ mathbb {R}}}$${\ displaystyle n \ in {\ mathbb {N}}}$${\ displaystyle p_ {n}}$${\ displaystyle p_ {n} (f): = \ sup _ {- n \ leq t \ leq n} | f (t) |}$${\ displaystyle C ({\ mathbb {R}})}$

## Inheritance properties

Subspaces , quotient spaces according to closed subspaces and countable products of metrizable, locally convex spaces are again of this type.

## Properties, distinguished spaces

Metrizable, locally convex spaces are bornological and therefore have all the properties of bornological spaces.

A bounded set in a topological vector space is a set B such that for every null neighborhood U there is a t> 0 with . This must not be confused with limitation in the metric sense. A metrically constrained quantity, i.e. H. a set with a finite diameter with respect to the metric, does not have to be restricted in the locally convex sense. ${\ displaystyle B \ subset tU}$

If E is a locally convex vector space, then every bounded set B in E defines a semi-norm on the dual space by setting. Provided with the set of semi-norms , where B runs through the bounded sets of E, becomes a locally convex vector space, which is then denoted by. ${\ displaystyle p_ {B}}$ ${\ displaystyle E \, '}$${\ displaystyle p_ {B} (f): = \ sup \ {| f (x) |; x \ in B \}}$${\ displaystyle p_ {B}}$${\ displaystyle E \, '}$${\ displaystyle E_ {b} '}$

For a metrizable, locally convex space are equivalent:

• ${\ displaystyle E_ {b} '}$is bornological .
• ${\ displaystyle E_ {b} '}$is quasitonneliert .
• ${\ displaystyle E_ {b} '}$is barreled .

A locally convex space that can be metrised is called distinguished if it fulfills these conditions. Standardized spaces, reflexive Fréchet spaces or quasi-standardizable Fréchet spaces are distinguished.

## swell

• K. Floret, J. Wloka: Introduction to the Theory of Locally Convex Spaces, Lecture Notes in Mathematics 56, 1968
• R. Meise, D. Vogt: Introduction to Functional Analysis, Vieweg, 1992 ISBN 3-528-07262-8