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 topological vector 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.
History
Metrizable topologies on vector spaces have been studied since their introduction in Maurice Fréchet's 1906 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,Hausdorff, F. Grundzüge der Mengenlehre (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.von Neumann, J. Collected works. Vol II. pp. 94–104Dieudonne, J. History of Functional Analysis Chapter VIII. Section 1. Finally, in 1935 von Neumann introduced the general definition of a locally convex space (called a convex space by him).von Neumann, J. Collected works. Vol II. pp. 508–527Dieudonne, J. History of Functional Analysis Chapter VIII. Section 2.
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 spacesBanach, S. Theory of linear operations p. 75. Ch. VIII. Sec. 3. Theorem 4., translated from Theorie des operations lineaires (1932) (in which case the unit ball of the dual is metrizable).
Definition
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.
Definition via convex sets
A topological vector space (TVS) is called if it has a neighborhood basis (that is, a local base) at the origin consisting of balanced, convex sets. The term is sometimes shortened to or .
In fact, every locally convex TVS has a neighborhood basis of the origin consisting of sets (that is, disks), where this neighborhood basis can further be chosen to also consist entirely of open sets or entirely of closed sets. Every TVS has a neighborhood basis at the origin consisting of balanced sets, but only a locally convex TVS has a neighborhood basis at the origin consisting of sets that are both balanced convex. It is possible for a TVS to have neighborhoods of the origin that are convex and yet not be locally convex because it has no neighborhood basis at the origin consisting entirely of convex sets (that is, every neighborhood basis at the origin contains some non-convex set); for example, every non-locally convex TVS has itself (that is, ) as a convex neighborhood of the origin.
Because translation is continuous (by definition of topological vector space), all translations are homeomorphisms, so every base for the neighborhoods of the origin can be translated to a base for the neighborhoods of any given vector.
Definition via seminorms
A seminorm on is a map such that
is nonnegative or positive semidefinite: ; is positive homogeneous or positive scalable: for every scalar So, in particular, ; is subadditive. It satisfies the triangle inequality:
If satisfies positive definiteness, which states that if then then is a norm. While seminorms need not be norms, there is an analogue of this criterion for families of seminorms—separatedness—defined below.
If is a vector space and is a family of seminorms on then a subset of is called a base of seminorms for if for all there exists a and a real such that
Definition (second version): A locally convex space is defined to be a vector space along with a family of seminorms on
Seminorm topology
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, making it into a topological vector space (TVS). By definition, it is the coarsest topology on for which all maps in are continuous.
It is possible for a locally convex topology on a space to be induced by a family of norms but for to be normable (that is, to have its topology be induced by a single norm).
Basis and subbases
An open set in has the form , where is a positive real number. The family of preimages as ranges over a family of seminorms and ranges over the positive real numbers is a subbasis at the origin for the topology induced by . These sets are convex, as follows from properties 2 and 3 of seminorms. Intersections of finitely many such sets are then also convex, and since the collection of all such finite intersections is a basis at the origin it follows that the topology is locally convex in the sense of the definition given above.
Recall that the topology of a TVS is translation invariant, meaning that if is any subset of containing the origin then for any is a neighborhood of the origin if and only if is a neighborhood of ; thus it suffices to define the topology at the origin. A base of neighborhoods of for this topology is obtained in the following way: for every finite subset of and every let
Bases of seminorms and saturated families
If is a locally convex space and if is a collection of continuous seminorms on , then is called a base of continuous seminorms if it is a base of seminorms for the collection of continuous seminorms on . Explicitly, this means that for all continuous seminorms on , there exists a and a real such that If is a base of continuous seminorms for a locally convex TVS then the family of all sets of the form as varies over and varies over the positive real numbers, is a of neighborhoods of the origin in (not just a subbasis, so there is no need to take finite intersections of such sets).Let be the open unit ball associated with the seminorm and note that if is real then and so Thus a basic open neighborhood of the origin induced by is a finite intersection of the form where and are all positive reals. Let which is a continuous seminorm and moreover, Pick and such that where this inequality holds if and only if Thus as desired.
A family of seminorms on a vector space is called saturated if for any and in the seminorm defined by belongs to
If is a saturated family of continuous seminorms that induces the topology on then the collection of all sets of the form as ranges over and ranges over all positive real numbers, forms a neighborhood basis at the origin consisting of convex open sets; This forms a basis at the origin rather than merely a subbasis so that in particular, there is need to take finite intersections of such sets.
Basis of norms
The following theorem implies that if is a locally convex space then the topology of can be a defined by a family of continuous on (a norm is a seminorm where implies ) if and only if there exists continuous 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).
Nets
Suppose that the topology of a locally convex space is induced by a family of continuous seminorms on . If and if is a net in , then in if and only if for all Moreover, if is Cauchy in , then so is for every
Equivalence of definitions
Although the definition in terms of a neighborhood base gives a better geometric picture, the definition in terms of seminorms is easier to work with in practice. The equivalence of the two definitions follows from a construction known as the Minkowski functional or Minkowski gauge. The key feature of seminorms which ensures the convexity of their -balls is the triangle inequality.
For an absorbing set such that if then whenever define the Minkowski functional of to be
From this definition it follows that is a seminorm if is balanced and convex (it is also absorbent by assumption). Conversely, given a family of seminorms, the sets
form a base of convex absorbent balanced sets.
Ways of defining a locally convex topology
Example: auxiliary normed spaces
If is convex and absorbing in then the symmetric set will be convex and balanced (also known as an or a ) in addition to being absorbing in This guarantees that the Minkowski functional of will be a seminorm on thereby making into a seminormed space that carries its canonical pseudometrizable topology. The set of scalar multiples as ranges over (or over any other set of non-zero scalars having as a limit point) forms a neighborhood basis of absorbing disks at the origin for this locally convex topology. If is a topological vector space and if this convex absorbing subset is also a bounded subset of then the absorbing disk will also be bounded, in which case will be a norm and will form what is known as an auxiliary normed space. If this normed space is a Banach space then is called a .
Further definitions
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.