Topological vector spaces
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 the complex numbers C or the real numbers R.
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) that is endowed with a topology such that vector addition X × X → X and scalar multiplication K × X → X are continuous functions (where the domains of these functions are endowed with product topologies).
Some authors (e.g., Rudin) require the topology on X to be T1; it then follows that the space is Hausdorff, and even Tychonoff. The topological and linear algebraic structures can be tied together even more closely with additional assumptions, the most common of which are listed below.
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