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 ${\displaystyle (U_{n})_{n\in {\mathbb {N} }}}$ ${\displaystyle U_{n}}$ ${\displaystyle n\in {\mathbb {N} }}$ ${\displaystyle p_{n}(x):=\inf\{t>0;x\in tU_{n}\}}$ ${\displaystyle E}$

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

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