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 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 formula_1 and every family formula_2 of topological "K"-vector spaces formula_3 and "K"-linear maps formula_4 there exists a vector space  topology formula_5 on formula_1 so that the following property is fulfilled: Whenever formula_7 is a "K"-linear map from a topological "K"-vector space formula_8 it holds that The topological vector space formula_12 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: