Category of topological vector spaces

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 TVect"K" 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.
TVectformula_1 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 formula_2 and every family formula_3 of topological "K"-vector spaces formula_4 and "K"-linear maps formula_5 there exists a vector space topology formula_6 on formula_2 so that the following property is fulfilled:
Whenever formula_8 is a "K"-linear map from a topological "K"-vector space formula_9 it holds that
The topological vector space formula_13 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: