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Concept
Topological vector space
Summary
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.
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.
There are topological vector spaces whose topology is not induced by a norm, but are still of interest in analysis.
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