Category of topological vector spaces

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In mathematics, the category of topological vector spaces is the category whose objects are topological vector spaces and whose morphisms are continuous linear maps between them. This is a category because the composition of two continuous linear maps is again a continuous linear map. The category is often denoted TVect or TVS.


Fixing a topological field K, one can also consider the subcategory TVectK of topological vector spaces over K with continuous K-linear maps as the morphisms.

TVect is a concrete category

Like many categories, the category TVect is a concrete category, meaning its objects are sets with additional structure (i.e. a vector space structure and a topology) and its morphisms are functions preserving this structure. There are obvious forgetful functors into the category of topological spaces, the category of vector spaces and the category of sets.

TVect is a topological category

The category is topological, which means loosely speaking that it relates to its "underlying category", the category of vector spaces, in the same way that Top relates to Set. Formally, for every K-vector space and every family of topological K-vector spaces and K-linear maps there exists a vector space topology on so that the following property is fulfilled:

Whenever is a K-linear map from a topological K-vector space it holds that

is continuous is continuous.

The topological vector space is called "initial object" or "initial structure" with respect to the given data.

If one replaces "vector space" by "set" and "linear map" by "map", one gets a characterisation of the usual initial topologies in Top. This is the reason why categories with this property are called "topological".

There are numerous consequences of this property. For example:

and the forgetful functor from to Set is right adjoint, the forgetful functor from to Top is right adjoint too (and the corresponding left adjoints fit in an analogue commutative diagram). This left adjoint defines "free topological vector spaces". Explicitly these are free K-vector spaces equipped with a certain initial topology.

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