Дөңес ашық жиындармен анықталған топологиялық векторлық кеңістіктер

A vector space with a topology defined by convex open sets
In functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological vector spaces whose topology is generated by translations of balanced, absorbent, convex sets. Alternatively they can be defined as a vector space with a family of seminorms, and a topology can be defined in terms of that family. Although in general such spaces are not necessarily normable, the existence of a convex local base for the zero vector is strong enough for the Hahn–Banach theorem to hold, yielding a sufficiently rich theory of continuous linear functionals. Fréchet spaces are locally convex spaces that are completely metrizable (with a choice of complete metric). They are generalizations of Banach spaces, which are complete vector spaces with respect to a metric generated by a norm.

Metrizable topologies on vector spaces have been studied since their introduction in Maurice Fréchet's 1902 PhD thesis Sur quelques points du calcul fonctionnel (wherein the notion of a metric was first introduced). After the notion of a general topological space was defined by Felix Hausdorff in 1914, although locally convex topologies were implicitly used by some mathematicians, up to 1934 only John von Neumann would seem to have explicitly defined the weak topology on Hilbert spaces and strong operator topology on operators on Hilbert spaces. Finally, in 1935 von Neumann introduced the general definition of a locally convex space (called a convex space by him). A notable example of a result which had to wait for the development and dissemination of general locally convex spaces (amongst other notions and results, like nets, the product topology and Tychonoff's theorem) to be proven in its full generality, is the Banach–Alaoglu theorem which Stefan Banach first established in 1932 by an elementary diagonal argument for the case of separable normed spaces (in which case the unit ball of the dual is metrizable).

Suppose is a vector space over a subfield of the complex numbers (normally itself or ). A locally convex space is defined either in terms of convex sets, or equivalently in terms of seminorms.

Suppose that is a vector space over where is either the real or complex numbers. A family of seminorms on the vector space induces a canonical vector space topology on , called the initial topology induced by the seminorms. By definition, it is the coarsest topology on for which all maps in are continuous. It will not carry the structure of a topological vector space (TVS) though. The vector space operations fail to be continuous in this topology, as all sets in the semi norm topology will be symmetric, where as the lcs topology allows for more sets. It is possible for a locally convex topology on a space to be induced by a family of norms but for to not be normable (that is, to have its topology be induced by a single norm).

The following theorem implies that if is a locally convex space then the topology of can be a defined by a family of continuous norms on (a norm is a seminorm where implies ) if and only if there exists at least one continuous norm on This is because the sum of a norm and a seminorm is a norm so if a locally convex space is defined by some family of seminorms (each of which is necessarily continuous) then the family of (also continuous) norms obtained by adding some given continuous norm to each element, will necessarily be a family of norms that defines this same locally convex topology. If there exists a continuous norm on a topological vector space then is necessarily Hausdorff but the converse is not in general true (not even for locally convex spaces or Fréchet spaces).

A family of seminorms is called total or separated or is said to separate points if whenever holds for every then is necessarily A locally convex space is Hausdorff if and only if it has a separated family of seminorms. Many authors take the Hausdorff criterion in the definition. A pseudometric is a generalization of a metric which does not satisfy the condition that only when A locally convex space is pseudometrizable, meaning that its topology arises from a pseudometric, if and only if it has a countable family of seminorms. Indeed, a pseudometric inducing the same topology is then given by (where the can be replaced by any positive summable sequence ). This pseudometric is translation invariant, but not homogeneous, meaning and therefore does not define a (pseudo)norm. The pseudometric is an honest metric if and only if the family of seminorms is separated, since this is the case if and only if the space is Hausdorff. If furthermore the space is complete, the space is called a Fréchet space. As with any topological vector space, a locally convex space is also a uniform space. Thus one may speak of uniform continuity, uniform convergence, and Cauchy sequences. A Cauchy net in a locally convex space is a net such that for every and every seminorm there exists some index such that for all indices In other words, the net must be Cauchy in all the seminorms simultaneously. The definition of completeness is given here in terms of nets instead of the more familiar sequences because unlike Fréchet spaces which are metrizable, general spaces may be defined by an uncountable family of pseudometrics. Sequences, which are countable by definition, cannot suffice to characterize convergence in such spaces. A locally convex space is complete if and only if every Cauchy net converges. A family of seminorms becomes a preordered set under the relation if and only if there exists an such that for all One says it is a directed family of seminorms if the family is a directed set with addition as the join, in other words if for every and there is a such that Every family of seminorms has an equivalent directed family, meaning one which defines the same topology. Indeed, given a family let be the set of finite subsets of and then for every define One may check that is an equivalent directed family. If the topology of the space is induced from a single seminorm, then the space is seminormable. Any locally convex space with a finite family of seminorms is seminormable. Moreover, if the space is Hausdorff (the family is separated), then the space is normable, with norm given by the sum of the seminorms. In terms of the open sets, a locally convex topological vector space is seminormable if and only if the origin has a bounded neighborhood.