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 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. Banach spaces, Hilbert spaces and Sobolev spaces are other well-known examples of TVSs.
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 or the real numbers unless clearly stated otherwise.
Motivation
Normed spaces
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: The vector addition map defined by is (jointly) continuous with respect to this topology. This follows directly from the triangle inequality obeyed by the norm. The scalar multiplication map defined by where is the underlying scalar field of is (jointly) continuous. This follows from the triangle inequality and homogeneity of the norm.
Thus all Banach spaces and Hilbert spaces are examples of topological vector spaces.
Non-normed 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.
Definition
A topological vector space (TVS) is a vector space over a topological field (most often the real or complex numbers with their standard topologies) that is endowed with a topology such that vector addition and scalar multiplication are continuous functions (where the domains of these functions are endowed with product topologies). Such a topology is called a or a on
Every topological vector space is also a commutative topological group under addition.
Hausdorff assumption
Many authors (for example, Walter Rudin), but not this page, require the topology on to be T1; it then follows that the space is Hausdorff, and even Tychonoff. A topological vector space is said to be if it is Hausdorff; importantly, "separated" does not mean separable. The topological and linear algebraic structures can be tied together even more closely with additional assumptions, the most common of which are listed below.
Category and morphisms
The category of topological vector spaces over a given topological field is commonly denoted or The objects are the topological vector spaces over and the morphisms are the continuous -linear maps from one object to another.
A (abbreviated ), also called a , is a continuous linear map between topological vector spaces (TVSs) such that the induced map is an open mapping when which is the range or image of is given the subspace topology induced by
A (abbreviated ), also called a , is an injective topological homomorphism. Equivalently, a TVS-embedding is a linear map that is also a topological embedding.
A (abbreviated ), also called a or an , is a bijective linear homeomorphism. Equivalently, it is a surjective TVS embedding
Many properties of TVSs that are studied, such as local convexity, metrizability, completeness, and normability, are invariant under TVS isomorphisms.
A necessary condition for a vector topology
A collection of subsets of a vector space is called if for every there exists some such that
All of the above conditions are consequently a necessity for a topology to form a vector topology.