Editing Topological vector space (section)

{{Short description|Vector space with a notion of nearness}}

In [[mathematics]], a '''topological vector space''' (also called a '''linear topological space''' and commonly abbreviated '''TVS''' or '''t.v.s.''') is one of the basic structures investigated in [[functional analysis]].
A topological vector space is a [[vector space]] that is also a [[topological space]] with the property that the vector space operations (vector addition and scalar multiplication) are also [[Continuous function|continuous functions]]. Such a topology is called a {{em|vector topology}} and every topological vector space has a [[Uniform space|uniform topological structure]], allowing a notion of [[uniform convergence]] and [[Complete topological vector space|completeness]]. Some authors also require that the space is a [[Hausdorff space]] (although this article does not). One of the most widely studied categories of TVSs are [[locally convex topological vector space]]s. This article focuses on TVSs that are not necessarily locally convex. Other well-known examples of TVSs include [[Banach space]]s, [[Hilbert space]]s and [[Sobolev space]]s.

Many topological vector spaces are spaces of [[Function (mathematics)|function]]s, or [[Linear map|linear operators]] acting on topological vector spaces, and the topology is often defined so as to capture a particular notion of [[Limit (mathematics)#Function space|convergence]] of sequences of functions.

In this article, the [[Scalar multiplication|scalar]] field of a topological vector space will be assumed to be either the [[complex number]]s <math>\Complex</math> or the [[real number]]s <math>\R,</math> unless clearly stated otherwise.