topological vector space
Definition
A topological vector space is a pair , where is a vector space over a topological field , and is a topology on such that under the scalar multiplication is a continuous function and the vector addition is a continuous function , where and are given the respective product topologies.
We will also require that is closed (which is equivalent to requiring the topology to be Hausdorff), though some authors do not make this requirement. Many authors require that be either or (with their usual topologies).
Topological vector spaces as topological groups
A topological vector space is necessarily a topological group: the definition ensures that the group operation (vector addition) is continuous, and the inverse operation is the same as multiplication by , and so is also continuous.
Finite-dimensional topological vector spaces
A finite-dimensional vector space inherits a natural topology. For if is a finite-dimensional vector space, then is isomorphic to for some ; then let be such an isomorphism, and suppose that has the product topology. Give the topology where a subset of is open in if and only if is open in . This topology is independent of the choice of isomorphism , and is the finest (http://planetmath.org/Coarser) topology on that makes it into a topological vector space.
Title | topological vector space |
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Canonical name | TopologicalVectorSpace |
Date of creation | 2013-03-22 12:16:55 |
Last modified on | 2013-03-22 12:16:55 |
Owner | yark (2760) |
Last modified by | yark (2760) |
Numerical id | 22 |
Author | yark (2760) |
Entry type | Definition |
Classification | msc 46A99 |
Synonym | TVS |
Synonym | linear topological space |
Synonym | topological linear space |
Related topic | TopologicalRing |
Related topic | FrechetSpace |