# Topological vector space

A **topological vector space** is a vector space on which not only its algebraic but also a compatible topological structure is defined.

## definition

Be . A - vector space , which also topological space is called
*a topological vector space* if the following compatibility axioms:

- The vector addition is continuous ,
- The scalar multiplication is continuous.

## Remarks

- It is important that the two maps mentioned are not only continuous in terms of components.
- Sometimes it is also required that a Kolmogoroff space (i.e. T
_{0}space) is, i.e. that different points are always topologically distinguishable. The Hausdorff property (i.e. T_{2}space) already follows for topological vector spaces ._{}_{} - is a topological group .
- For a topological vector space , the topological dual space can be explained in a meaningful way .
- If the topological vector space is a Hausdorff space, then the mappings that represent a shift by a certain vector or a stretching by a scalar are homeomorphisms . In this case it is sufficient to consider topological properties of the space in the origin, since any amount can be moved homeomorphically into the origin.

## Examples

- The most important examples are the normalized vector spaces , including the Banach spaces . Important concrete examples here are the Euclidean vector space , the spaces (with ) and Sobolev spaces .

- More general examples are the locally convex spaces , including the Fréchet spaces . Important concrete examples here are the areas of distribution theory , ie , , , , and .

- The set is a vector space that becomes a topological vector space for the metric that is not locally convex.

- More general are a measure space and . Then the metric makes the L
^{p}space a topological vector space that is generally not locally convex. If and is the counting measure , the above example is obtained . Apart from the zero functional, space has no other continuous linear functional .^{}

- Each vector space has the chaotic topology , that is, only the empty set and the entire space are open , a topological vector space.

## Topological properties

- As an Abelian, topological group, every topological vector space is a uniform space . In particular, it is always an R
_{0}space and fulfills the axiom of separation T_{3}(meaning that T_{0 is}not included). This uniform structure can be used to define completeness and uniform continuity . Any topological vector space can be completed and linear continuous mappings between topological vector spaces are uniformly continuous. - For a topological vector space is considered: is T
_{0}is T_{1}is T_{2}is a Tychonoff space ._{}_{}_{} - Every topological vector space has a zero neighborhood basis made up of closed and balanced sets . According to Kolmogoroff's criterion for normalization , a Hausdorff topological vector space can be normalized if and only if it has a bounded and convex null neighborhood.
- In locally convex Hausdorff topological vector spaces which applies Hahn-Banach theorem , so the existence of "many" continuous linear functional is secured. This fact makes it possible to set up a rich duality theory for such spaces , which does not apply to general topological vector spaces in this form. In the extreme case, as in the example above , the zero functional is the only continuous linear functional.

## See also

## literature

- Helmut H. Schaefer:
*Topological Vector Spaces*. Springer, New York et al. 1971, ISBN 0-387-98726-6 . - Hans Jarchow:
*Locally Convex Spaces*. Teubner, Stuttgart 1981, ISBN 3-519-02224-9 .