Topological Vector Space

A topological vector space X is a vector space over a topological field K (most often the real or complex numbers with their standard topologies) which is endowed with a topology such that vector addition X × X → X and scalar multiplication K × X → X are continuous functions (where the domains of these functions are endowed with product topologies).

Some authors (e.g., Rudin) require the topology on X to be T₀; it then follows that the space is Hausdorff, and even Tychonoff. The topological and linear algebraic structures can be tied together even more closely with additional assumptions, the most common of which are listed below.

The category of topological vector spaces over a given topological field K is commonly denoted TVS_K or TVect_K. The objects are the topological vector spaces over K and the morphisms are the continuous K-linear maps from one object to another.