Topological vector space

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


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


  • 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.


  • 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 .

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


  • 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 .