Topological vector space

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In mathematics, a topological vector space (also called a linear topological space) is one of the basic structures investigated in functional analysis. As the name suggests the space blends a topological structure (a uniform structure to be precise) with the algebraic concept of a vector space.

The elements of topological vector spaces are typically functions or linear operators acting on topological vector spaces, and the topology is often defined so as to capture a particular notion of convergence of sequences of functions.

Hilbert spaces and Banach spaces are well-known examples.

Unless stated otherwise, the underlying field of a topological vector space is assumed to be either $\mathbf C$ or $\mathbf R$.

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Definition

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 × XX and scalar multiplication K × XX are continuous functions.

Some authors (e.g., Rudin) require the topology on X to be Hausdorff, and some additionally require the topology on X to be locally convex (e.g., Fréchet space). For a topological vector space to be Hausdorff it suffices that the space be T0; it then follows that the space is even T3½ .

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