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 continuous. The category is often denoted TVect or TVS.

Fixing a topological field K, one can also consider the (sub-)category TVect_K of topological vector spaces over K with continuous K-linear maps as the morphisms.

[edit] 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.