structural:reformatting, linguistic:paraphrasing
Topological vector spaces

Convergence questions are treated by considering vector spaces V {\displaystyle V} carrying a compatible topology, a structure that allows one to talk about elements being close to each other. Compatible here means that addition and scalar multiplication have to be continuous maps. Roughly, if x {\displaystyle \mathbf {x} } and y {\displaystyle \mathbf {y} } in V {\displaystyle V}, and a {\displaystyle a} in F {\displaystyle F} vary by a bounded amount, then so do x + y {\displaystyle \mathbf {x} +\mathbf {y} } and a x. {\displaystyle a\mathbf {x}.} To make sense of specifying the amount a scalar changes, the field F {\displaystyle F} also has to carry a topology in this context; a common choice is the reals or the complex numbers.

In such topological vector spaces one can consider series of vectors. The infinite sum ∑ i = 1 ∞ f i = lim n → ∞ f 1 + ⋯ + f n {\displaystyle \sum _{i=1}^{\infty }f_{i}~=~\lim _{n\to \infty }f_{1}+\cdots +f_{n}} denotes the limit of the corresponding finite partial sums of the sequence f 1, f 2, … {\displaystyle f_{1},f_{2},\ldots } of elements of V. {\displaystyle V.} For example, the f i {\displaystyle f_{i}} could be (real or complex) functions belonging to some function space V, {\displaystyle V,} in which case the series is a function series. The mode of convergence of the series depends on the topology imposed on the function space. In such cases, pointwise convergence and uniform convergence are two prominent examples.

A way to ensure the existence of limits of certain infinite series is to restrict attention to spaces where any Cauchy sequence has a limit; such a vector space is called complete. Roughly, a vector space is complete provided that it contains all necessary limits. For example, the vector space of polynomials on the unit interval, {\displaystyle,} equipped with the topology of uniform convergence is not complete because any continuous function on {\displaystyle } can be uniformly approximated by a sequence of polynomials, by the Weierstrass approximation theorem. In contrast, the space of all continuous functions on {\displaystyle } with the same topology is complete. A norm gives rise to a topology by defining that a sequence of vectors v n {\displaystyle \mathbf {v} _{n}} converges to v {\displaystyle \mathbf {v} } if and only if lim n → ∞ | v n − v | = 0. {\displaystyle \lim _{n\to \infty }|\mathbf {v} _{n}-\mathbf {v} |=0.} Banach and Hilbert spaces are complete topological vector spaces whose topologies are given, respectively, by a norm and an inner product. Their study—a key piece of functional analysis —focuses on infinite-dimensional vector spaces, since all norms on finite-dimensional topological vector spaces give rise to the same notion of convergence. The image at the right shows the equivalence of the 1 {\displaystyle 1} -norm and ∞ {\displaystyle \infty } -norm on R 2: {\displaystyle \mathbf {R} ^{2}:} as the unit "balls" enclose each other, a sequence converges to zero in one norm if and only if it so does in the other norm. In the infinite-dimensional case, however, there will generally be inequivalent topologies, which makes the study of topological vector spaces richer than that of vector spaces without additional data.

From a conceptual point of view, all notions related to topological vector spaces should match the topology. For example, instead of considering all linear maps (also called functionals) V → W, {\displaystyle V\to W,} maps between topological vector spaces are required to be continuous. In particular, the (topological) dual space V ∗ {\displaystyle V^{*}} consists of continuous functionals V → R {\displaystyle V\to \mathbf {R} } (or to C {\displaystyle \mathbf {C} }). The fundamental Hahn–Banach theorem is concerned with separating subspaces of appropriate topological vector spaces by continuous functionals.