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 functions. Such a topology is called a vector topology and every topological vector space has a uniform topological structure, allowing a notion of uniform convergence and 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 spaces. This article focuses on TVSs that are not necessarily locally convex. Other well-known examples of TVSs include Banach spaces, Hilbert spaces and Sobolev spaces.

Many topological vector spaces are spaces of 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.

In this article, the scalar field of a topological vector space will be assumed to be either the complex numbers ${\displaystyle \mathbb {C} }$ or the real numbers ${\displaystyle \mathbb {R} ,}$ unless clearly stated otherwise.

Every normed vector space has a natural topological structure: the norm induces a metric and the metric induces a topology. This is a topological vector space because^{[citation needed]}:

Thus all Banach spaces and Hilbert spaces are examples of topological vector spaces.

There are topological vector spaces whose topology is not induced by a norm, but are still of interest in analysis. Examples of such spaces are spaces of holomorphic functions on an open domain, spaces of infinitely differentiable functions, the Schwartz spaces, and spaces of test functions and the spaces of distributions on them. These are all examples of Montel spaces. An infinite-dimensional Montel space is never normable. The existence of a norm for a given topological vector space is characterized by Kolmogorov's normability criterion.

A topological field is a topological vector space over each of its subfields.

A topological vector space (TVS) ${\displaystyle X}$ is a vector space over a topological field ${\displaystyle \mathbb {K} }$ (most often the real or complex numbers with their standard topologies) that is endowed with a topology such that vector addition ${\displaystyle \cdot \,+\,\cdot \;:X\times X\to X}$ and scalar multiplication ${\displaystyle \cdot :\mathbb {K} \times X\to X}$ are continuous functions (where the domains of these functions are endowed with product topologies). Such a topology is called a vector topology or a TVS topology on ${\displaystyle X.}$