topological vector space
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.
Topological vector spaces as topological groups
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|
|Date of creation||2013-03-22 12:16:55|
|Last modified on||2013-03-22 12:16:55|
|Last modified by||yark (2760)|
|Synonym||linear topological space|
|Synonym||topological linear space|