# topological vector space

## Definition

A *topological vector space ^{}* is a pair $(V,\mathcal{T})$,
where $V$ is a vector space

^{}over a topological field $K$, and $\mathcal{T}$ is a topology

^{}on $V$ such that under $\mathcal{T}$ the scalar multiplication $(\lambda ,v)\mapsto \lambda v$ is a continuous function

^{}$K\times V\to V$ and the vector addition $(v,w)\mapsto v+w$ is a continuous function $V\times V\to V$, where $K\times V$ and $V\times V$ are given the respective product topologies.

We will also require that $\{0\}$ is closed
(which is equivalent^{} to requiring the topology to be Hausdorff^{}),
though some authors do not make this requirement.
Many authors require that $K$ be either $\mathbb{R}$ or $\u2102$
(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 $-1$,
and so is also continuous.

## Finite-dimensional topological vector spaces

A finite-dimensional vector space inherits a natural topology.
For if $V$ is a finite-dimensional vector space,
then $V$ is isomorphic^{} to ${K}^{n}$ for some $n$;
then let $f:V\to {K}^{n}$ be such an isomorphism^{},
and suppose that ${K}^{n}$ has the product topology.
Give $V$ the topology where a subset $A$ of $V$ is open in $V$
if and only if $f(A)$ is open in ${K}^{n}$.
This topology is independent of the choice of isomorphism $f$,
and is the finest (http://planetmath.org/Coarser^{}) topology on $V$
that makes it into a topological vector space.

Title | topological vector space |
---|---|

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 |