Topological vector space

In mathematics, a topological vector space (also called a linear topological space) is one of the basic structures investigated in functional analysis. A topological vector space is a vector space (an algebraic structure) which is also a topological space, thereby admitting a notion of continuity. More specifically, its topological space has a uniform topological structure, allowing a notion of uniform convergence. In mathematics, a topological vector space (also called a linear topological space) is one of the basic structures investigated in functional analysis. A topological vector space is a vector space (an algebraic structure) which is also a topological space, thereby admitting a notion of continuity. More specifically, its topological space has a uniform topological structure, allowing a notion of uniform convergence. 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., Walter 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. The category of topological vector spaces over a given topological field K is commonly denoted TVSK or TVectK. The objects are the topological vector spaces over K and the morphisms are the continuous K-linear maps from one object to another. 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: Therefore, all Banach spaces and Hilbert spaces are examples of topological vector spaces.