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 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).

Definition

Suppose X is a vector space over \mathbb{K}, a subfield of the complex numbers (normally \Complex itself or \R). 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. A subset C in X is called 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 X has itself (that is, X) 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 X is a map such that If p satisfies positive definiteness, which states that if p(x) = 0 then x = 0, then p is a norm. While in general seminorms need not be norms, there is an analogue of this criterion for families of seminorms, separatedness, defined below. If X is a vector space and \mathcal{P} is a family of seminorms on X then a subset \mathcal{Q} of \mathcal{P} is called a base of seminorms for \mathcal{P} if for all there exists a and a real r > 0 such that p \leq r q. Definition (second version): A locally convex space is defined to be a vector space X along with a family \mathcal{P} of seminorms on X.

Seminorm topology

Suppose that X is a vector space over \mathbb{K}, where \mathbb{K} is either the real or complex numbers. A family of seminorms \mathcal{P} on the vector space X induces a canonical vector space topology on X, called the initial topology induced by the seminorms, making it into a topological vector space (TVS). By definition, it is the coarsest topology on X for which all maps in \mathcal{P} are continuous. It is possible for a locally convex topology on a space X to be induced by a family of norms but for X 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 [0, r), where r is a positive real number. The family of preimages as p ranges over a family of seminorms \mathcal{P} and r ranges over the positive real numbers is a subbasis at the origin for the topology induced by \mathcal{P}. 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 S is any subset of X containing the origin then for any x \in X, S is a neighborhood of the origin if and only if x + S is a neighborhood of x; thus it suffices to define the topology at the origin. A base of neighborhoods of y for this topology is obtained in the following way: for every finite subset F of \mathcal{P} and every r > 0, let

Bases of seminorms and saturated families

If X is a locally convex space and if \mathcal{P} is a collection of continuous seminorms on X, then \mathcal{P} is called a base of continuous seminorms if it is a base of seminorms for the collection of continuous seminorms on X. Explicitly, this means that for all continuous seminorms p on X, there exists a and a real r > 0 such that p \leq r q. If \mathcal{P} is a base of continuous seminorms for a locally convex TVS X then the family of all sets of the form as q varies over \mathcal{P} and r varies over the positive real numbers, is a of neighborhoods of the origin in X (not just a subbasis, so there is no need to take finite intersections of such sets). A family \mathcal{P} of seminorms on a vector space X is called saturated if for any p and q in the seminorm defined by belongs to If \mathcal{P} is a saturated family of continuous seminorms that induces the topology on X then the collection of all sets of the form as p ranges over \mathcal{P} and r 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 X is a locally convex space then the topology of X can be a defined by a family of continuous on X (a norm is a semi**norm** s where s(x)=0 implies x=0) if and only if there exists continuous on X. This is because the sum of a norm and a semi**norm** is a norm so if a locally convex space is defined by some family \mathcal{P} of semi**norm**s (each of which is necessarily continuous) then the family of (also continuous) **norm**s obtained by adding some given continuous norm n to each element, will necessarily be a family of **norm**s that defines this same locally convex topology. If there exists a continuous norm on a topological vector space X then X 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 X is induced by a family \mathcal{P} of continuous seminorms on X. If x \in X and if is a net in X, then in X if and only if for all Moreover, if x_{\bull} is Cauchy in X, 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 \varepsilon-balls is the triangle inequality. For an absorbing set C such that if x \in C, then t x \in C whenever define the Minkowski functional of C to be From this definition it follows that \mu_C is a seminorm if C 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 W is convex and absorbing in X then the symmetric set will be convex and balanced (also known as an or a ) in addition to being absorbing in X. This guarantees that the Minkowski functional of D will be a seminorm on X, thereby making into a seminormed space that carries its canonical pseudometrizable topology. The set of scalar multiples r D as r ranges over (or over any other set of non-zero scalars having 0 as a limit point) forms a neighborhood basis of absorbing disks at the origin for this locally convex topology. If X is a topological vector space and if this convex absorbing subset W is also a bounded subset of X, then the absorbing disk will also be bounded, in which case p_D will be a norm and will form what is known as an auxiliary normed space. If this normed space is a Banach space then D is called a.

Further definitions

Sufficient conditions

Hahn–Banach extension property

Let X be a TVS. Say that a vector subspace M of X has the extension property if any continuous linear functional on M can be extended to a continuous linear functional on X. Say that X has the Hahn-Banach extension property (HBEP) if every vector subspace of X has the extension property. The Hahn-Banach theorem guarantees that every Hausdorff locally convex space has the HBEP. For complete metrizable TVSs there is a converse: If a vector space X has uncountable dimension and if we endow it with the finest vector topology then this is a TVS with the HBEP that is neither locally convex or metrizable.

Properties

Throughout, \mathcal{P} is a family of continuous seminorms that generate the topology of X. Topological closure If and x \in X, then if and only if for every r > 0 and every finite collection there exists some s \in S such that The closure of {0} in X is equal to Topology of Hausdorff locally convex spaces Every Hausdorff locally convex space is homeomorphic to a vector subspace of a product of Banach spaces. The Anderson–Kadec theorem states that every infinite–dimensional separable Fréchet space is homeomorphic to the product space of countably many copies of \R (this homeomorphism need not be a linear map).

Properties of convex subsets

Algebraic properties of convex subsets A subset C is convex if and only if for all or equivalently, if and only if for all positive real where because always holds, the equals sign ,=, can be replaced with If C is a convex set that contains the origin then C is star shaped at the origin and for all non-negative real The Minkowski sum of two convex sets is convex; furthermore, the scalar multiple of a convex set is again convex. Topological properties of convex subsets

Properties of convex hulls

For any subset S of a TVS X, the convex hull (respectively, closed convex hull, balanced hull, convex balanced hull) of S, denoted by (respectively, ), is the smallest convex (respectively, closed convex, balanced, convex balanced) subset of X containing S.

Examples and nonexamples

Finest and coarsest locally convex topology

Coarsest vector topology

Any vector space X endowed with the trivial topology (also called the indiscrete topology) is a locally convex TVS (and of course, it is the coarsest such topology). This topology is Hausdorff if and only X = {0}. The indiscrete topology makes any vector space into a complete pseudometrizable locally convex TVS. In contrast, the discrete topology forms a vector topology on X if and only X = {0}. This follows from the fact that every topological vector space is a connected space.

Finest locally convex topology

If**** X is**** a real**** or**** complex vector**** space and if**** *mat**hc**al**{P}*** is**** the set of**** all seminorms on**** X then**** the locally convex**** TVS topology****,**** denoted by**** that**** *mat**hc**al**{P}*** induces on**** X is**** called**** the **** on**** X.**** This**** topology**** may also**** be**** described as**** the TVS-topology**** on**** X having**** as**** a neighborhood**** base**** at**** the origin**** the set of**** all absorbing disks in**** X.**** Any locally convex**** TVS-topology**** on**** X is**** necessarily a subset**** of**** **** **** is**** Hausdorff.**** Every linear map from into another locally convex TVS is necessarily continuous. In particular, every linear functional on is continuous and every vector subspace of X is closed in ; therefore, if X is infinite dimensional then is not pseudometrizable (and thus not metrizable). Moreover, is the Hausdorff locally convex topology on X with the property that any linear map from it into any Hausdorff locally convex space is continuous. The space is a bornological space.

Examples of locally convex spaces

Every normed space is a Hausdorff locally convex space, and much of the theory of locally convex spaces generalizes parts of the theory of normed spaces. The family of seminorms can be taken to be the single norm. Every Banach space is a complete Hausdorff locally convex space, in particular, the L^p spaces with p \geq 1 are locally convex. More generally, every Fréchet space is locally convex. A Fréchet space can be defined as a complete locally convex space with a separated countable family of seminorms. The space \R^{\omega} of real valued sequences with the family of seminorms given by is locally convex. The countable family of seminorms is complete and separable, so this is a Fréchet space, which is not normable. This is also the limit topology of the spaces \R^n, embedded in \R^{\omega} in the natural way, by completing finite sequences with infinitely many 0. Given any vector space X and a collection F of linear functionals on it, X can be made into a locally convex topological vector space by giving it the weakest topology making all linear functionals in F continuous. This is known as the weak topology or the initial topology determined by F. The collection F may be the algebraic dual of X or any other collection. The family of seminorms in this case is given by for all f in F. Spaces of differentiable functions give other non-normable examples. Consider the space of smooth functions such that where a and B are multiindices. The family of seminorms defined by is separated, and countable, and the space is complete, so this metrizable space is a Fréchet space. It is known as the Schwartz space, or the space of functions of rapid decrease, and its dual space is the space of tempered distributions. An important function space in functional analysis is the space D(U) of smooth functions with compact support in A more detailed construction is needed for the topology of this space because the space is not complete in the uniform norm. The topology on D(U) is defined as follows: for any fixed compact set the space of functions with is a Fréchet space with countable family of seminorms (these are actually norms, and the completion of the space with the norm is a Banach space D^m(K)). Given any collection of compact sets, directed by inclusion and such that their union equal U, the form a direct system, and D(U) is defined to be the limit of this system. Such a limit of Fréchet spaces is known as an LF space. More concretely, D(U) is the union of all the with the strongest topology which makes each inclusion map continuous. This space is locally convex and complete. However, it is not metrizable, and so it is not a Fréchet space. The dual space of is the space of distributions on \R^n. More abstractly, given a topological space X, the space C(X) of continuous (not necessarily bounded) functions on X can be given the topology of uniform convergence on compact sets. This topology is defined by semi-norms (as K varies over the directed set of all compact subsets of X). When X is locally compact (for example, an open set in \R^n) the Stone–Weierstrass theorem applies—in the case of real-valued functions, any subalgebra of C(X) that separates points and contains the constant functions (for example, the subalgebra of polynomials) is dense.

Examples of spaces lacking local convexity

Many topological vector spaces are locally convex. Examples of spaces that lack local convexity include the following: Both examples have the property that any continuous linear map to the real numbers is 0. In particular, their dual space is trivial, that is, it contains only the zero functional.

Continuous mappings

Because locally convex spaces are topological spaces as well as vector spaces, the natural functions to consider between two locally convex spaces are continuous linear maps. Using the seminorms, a necessary and sufficient criterion for the continuity of a linear map can be given that closely resembles the more familiar boundedness condition found for Banach spaces. Given locally convex spaces X and Y with families of seminorms and respectively, a linear map T : X \to Y is continuous if and only if for every \beta, there exist and M > 0 such that for all v \in X, In other words, each seminorm of the range of T is bounded above by some finite sum of seminorms in the domain. If the family is a directed family, and it can always be chosen to be directed as explained above, then the formula becomes even simpler and more familiar: The class of all locally convex topological vector spaces forms a category with continuous linear maps as morphisms.

Linear functionals

If X is a real or complex vector space, f is a linear functional on X, and p is a seminorm on X, then |f| \leq p if and only if f \leq p. If f is a non-0 linear functional on a real vector space X and if p is a seminorm on X, then f \leq p if and only if

Multilinear maps

Let n \geq 1 be an integer, be TVSs (not necessarily locally convex), let Y be a locally convex TVS whose topology is determined by a family \mathcal{Q} of continuous seminorms, and let be a multilinear operator that is linear in each of its n coordinates. The following are equivalent: