Topological vector spaces are the most general spaces investigated in functional analysis. The elements of topological vector spaces are typically functions, and the topology is often defined so as to capture a particular notion of convergence of sequences of functions.
All normed vector spaces (and therefore all Banach spaces and Hilbert spaces) are examples of topological vector spaces. There are also many other examples however.
Consider for instance the set X of all functions f : R → R. X can be identified with the product space R^{R} and carries a natural product topology. With this topology, X becomes a topological vector space, called the space of pointwise convergence. The reason for this name is the following: if (f_{n}) is a sequence of elements in X, then f_{n} has limit f in X if and only if f_{n}(x) has limit f(x) for every real number x.
Here is another example: consider an open set D in R^{n} and the set X of infinitely differentiable functions f : D → R. We first define a collection of seminorms on X, and the topology will then be defined as the coarsest topology which refines the topology defined by each of the seminorms. For a compact set K and a multiindex m = (m_{1}, ..., m_{n}) we define the (K, m) seminorm to be the supremum of the differentiation first by x_{1} m_{1} times, then by x_{2} m_{2} times and so on K. With this topology, a sequence (f_{n}) in X has limit f if and only if on every compact set all derivatives of f_{n} converge uniformly to the corresponding derivative of f.
Types of topological vector spaces
We start the list with the most general classes and proceed to the "nicer" ones.
The negation in every topological vector space is continuous (since it is the same as multiplication by 1), so every topological vector space is a topological group. In particular, topological vector spaces are uniform spaces and one can thus talk about completeness, uniform convergence and uniform continuity. The vector space operations of addition and scalar multiplication are actually uniformly continuous. Because of this, every topological vector space can be completed and is thus a dense linear subspace of a complete topological vector space.
A topological vector space is finitedimensional if and only if it is locally compact, in which case it is isomorphic to a Euclidean space R^{n} or C^{n} (in the sense that there exists a linear homeomorphism between the two spaces).
Every topological vector space has a dual  the set V^{*} of all continuous functionals, i.e. continuous linear maps from the space into the base field K. The topology on the dual can be defined to be the coarsest topology such that the appliance mapping V × V^{*} > K is continuous. This turns the dual into a topological vector space. Note that in the case of a Banach space, this gives the dual the weak* topology
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