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.
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, [1] 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. [2] [3] Finally, in 1935 von Neumann introduced the general definition of a locally convex space (called a convex space by him). [4] [5]
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 [6] (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.
A topological vector space (TVS) is called locally convex if it has a neighborhood basis (that is, a local base) at the origin consisting of balanced, convex sets. [7] The term locally convex topological vector space is sometimes shortened to locally convex space or LCTVS.
A subset in is called
In fact, every locally convex TVS has a neighborhood basis of the origin consisting of absolutely convex sets (that is, disks), where this neighborhood basis can further be chosen to also consist entirely of open sets or entirely of closed sets. [8] 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 and convex. It is possible for a TVS to have some 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.
A seminorm on is a map such that
If satisfies positive definiteness, which states that if then then 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 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 [9]
Definition (second version): A locally convex space is defined to be a vector space along with a family of seminorms on
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 not be normable (that is, to have its topology be induced by a single norm).
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 first 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
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 all continuous seminorms on . [9] Explicitly, this means that for all continuous seminorms on , there exists a and a real such that [9] 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 base of neighborhoods of the origin in (not just a subbasis, so there is no need to take finite intersections of such sets). [9] [proof 1]
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; [9] This forms a basis at the origin rather than merely a subbasis so that in particular, there is no need to take finite intersections of such sets. [9]
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 . [10] 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).
Theorem [11] — Let be a Fréchet space over the field Then the following are equivalent:
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 [12] Moreover, if is Cauchy in , then so is for every [12]
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.
Theorem [7] — Suppose that is a (real or complex) vector space and let be a filter base of subsets of such that:
Then is a neighborhood base at 0 for a locally convex TVS topology on
Theorem [7] — Suppose that is a (real or complex) vector space and let be a non-empty collection of convex, balanced, and absorbing subsets of Then the set of all positive scalar multiples of finite intersections of sets in forms a neighborhood base at the origin for a locally convex TVS topology on
Example: auxiliary normed spaces
If is convex and absorbing in then the symmetric set will be convex and balanced (also known as an absolutely convex set or a disk ) 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 Banach disk .
Let be a TVS. Say that a vector subspace of has the extension property if any continuous linear functional on can be extended to a continuous linear functional on . [13] Say that has the Hahn-Banach extension property (HBEP) if every vector subspace of has the extension property. [13]
The Hahn-Banach theorem guarantees that every Hausdorff locally convex space has the HBEP. For complete metrizable TVSs there is a converse:
Theorem [13] (Kalton) — Every complete metrizable TVS with the Hahn-Banach extension property is locally convex.
If a vector space 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. [13]
Throughout, is a family of continuous seminorms that generate the topology of
Topological closure
If and then if and only if for every and every finite collection there exists some such that [14] The closure of in is equal to [15]
Topology of Hausdorff locally convex spaces
Every Hausdorff locally convex space is homeomorphic to a vector subspace of a product of Banach spaces. [16] The Anderson–Kadec theorem states that every infinite–dimensional separable Fréchet space is homeomorphic to the product space of countably many copies of (this homeomorphism need not be a linear map). [17]
Algebraic properties of convex subsets
A subset is convex if and only if for all [18] or equivalently, if and only if for all positive real [19] where because always holds, the equals sign can be replaced with If is a convex set that contains the origin then 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. [20]
Topological properties of convex subsets
For any subset of a TVS the convex hull (respectively, closed convex hull, balanced hull , convex balanced hull) of denoted by (respectively, ), is the smallest convex (respectively, closed convex, balanced, convex balanced) subset of containing
Any vector space 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 The indiscrete topology makes any vector space into a complete pseudometrizable locally convex TVS.
In contrast, the discrete topology forms a vector topology on if and only This follows from the fact that every topological vector space is a connected space.
If is a real or complex vector space and if is the set of all seminorms on then the locally convex TVS topology, denoted by that induces on is called the finest locally convex topology on [37] This topology may also be described as the TVS-topology on having as a neighborhood base at the origin the set of all absorbing disks in [37] Any locally convex TVS-topology on is necessarily a subset of is Hausdorff. [15] Every linear map from into another locally convex TVS is necessarily continuous. [15] In particular, every linear functional on is continuous and every vector subspace of is closed in ; [15] therefore, if is infinite dimensional then is not pseudometrizable (and thus not metrizable). [37] Moreover, is the only Hausdorff locally convex topology on with the property that any linear map from it into any Hausdorff locally convex space is continuous. [38] The space is a bornological space. [39]
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 spaces with 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 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 embedded in in the natural way, by completing finite sequences with infinitely many
Given any vector space and a collection of linear functionals on it, can be made into a locally convex topological vector space by giving it the weakest topology making all linear functionals in continuous. This is known as the weak topology or the initial topology determined by The collection may be the algebraic dual of or any other collection. The family of seminorms in this case is given by for all in
Spaces of differentiable functions give other non-normable examples. Consider the space of smooth functions such that where and 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 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 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 ). Given any collection of compact sets, directed by inclusion and such that their union equal the form a direct system, and is defined to be the limit of this system. Such a limit of Fréchet spaces is known as an LF space. More concretely, is the union of all the with the strongest locally convex 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
More abstractly, given a topological space the space of continuous (not necessarily bounded) functions on can be given the topology of uniform convergence on compact sets. This topology is defined by semi-norms (as varies over the directed set of all compact subsets of ). When is locally compact (for example, an open set in ) the Stone–Weierstrass theorem applies—in the case of real-valued functions, any subalgebra of that separates points and contains the constant functions (for example, the subalgebra of polynomials) is dense.
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 In particular, their dual space is trivial, that is, it contains only the zero functional.
Theorem [40] — Let be a linear operator between TVSs where is locally convex (note that need not be locally convex). Then is continuous if and only if for every continuous seminorm on , there exists a continuous seminorm on such that
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 and with families of seminorms and respectively, a linear map is continuous if and only if for every there exist and such that for all
In other words, each seminorm of the range of 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.
Theorem [40] — If is a TVS (not necessarily locally convex) and if is a linear functional on , then is continuous if and only if there exists a continuous seminorm on such that
If is a real or complex vector space, is a linear functional on , and is a seminorm on , then if and only if [41] If is a non-0 linear functional on a real vector space and if is a seminorm on , then if and only if [15]
Let be an integer, be TVSs (not necessarily locally convex), let be a locally convex TVS whose topology is determined by a family of continuous seminorms, and let be a multilinear operator that is linear in each of its coordinates. The following are equivalent:
The Hahn–Banach theorem is a central tool in functional analysis. It allows the extension of bounded linear functionals defined on a vector subspace of some vector space to the whole space, and it also shows that there are "enough" continuous linear functionals defined on every normed vector space to make the study of the dual space "interesting". Another version of the Hahn–Banach theorem is known as the Hahn–Banach separation theorem or the hyperplane separation theorem, and has numerous uses in convex geometry.
In mathematics, a topological vector space is one of the basic structures investigated in functional analysis. A topological vector space is a vector space that is also a topological space with the property that the vector space operations are also continuous functions. Such a topology is called a vector topology and every topological vector space has a uniform topological structure, allowing a notion of uniform convergence and completeness. Some authors also require that the space is a Hausdorff space. One of the most widely studied categories of TVSs are locally convex topological vector spaces. This article focuses on TVSs that are not necessarily locally convex. Other well-known examples of TVSs include Banach spaces, Hilbert spaces and Sobolev spaces.
In functional analysis and related areas of mathematics, Fréchet spaces, named after Maurice Fréchet, are special topological vector spaces. They are generalizations of Banach spaces. All Banach and Hilbert spaces are Fréchet spaces. Spaces of infinitely differentiable functions are typical examples of Fréchet spaces, many of which are typically not Banach spaces.
In mathematics, particularly in functional analysis, a seminorm is a norm that need not be positive definite. Seminorms are intimately connected with convex sets: every seminorm is the Minkowski functional of some absorbing disk and, conversely, the Minkowski functional of any such set is a seminorm.
In functional analysis and related areas of mathematics, a barrelled space is a topological vector space (TVS) for which every barrelled set in the space is a neighbourhood for the zero vector. A barrelled set or a barrel in a topological vector space is a set that is convex, balanced, absorbing, and closed. Barrelled spaces are studied because a form of the Banach–Steinhaus theorem still holds for them. Barrelled spaces were introduced by Bourbaki.
In functional and convex analysis, and related disciplines of mathematics, the polar set is a special convex set associated to any subset of a vector space lying in the dual space The bipolar of a subset is the polar of but lies in .
In functional analysis and related areas of mathematics, a set in a topological vector space is called bounded or von Neumann bounded, if every neighborhood of the zero vector can be inflated to include the set. A set that is not bounded is called unbounded.
In functional analysis and related areas of mathematics a polar topology, topology of -convergence or topology of uniform convergence on the sets of is a method to define locally convex topologies on the vector spaces of a pairing.
In functional analysis and related areas of mathematics, a continuous linear operator or continuous linear mapping is a continuous linear transformation between topological vector spaces.
In mathematics, particularly in functional analysis, a bornological space is a type of space which, in some sense, possesses the minimum amount of structure needed to address questions of boundedness of sets and linear maps, in the same way that a topological space possesses the minimum amount of structure needed to address questions of continuity. Bornological spaces are distinguished by the property that a linear map from a bornological space into any locally convex spaces is continuous if and only if it is a bounded linear operator.
In mathematics, an LF-space, also written (LF)-space, is a topological vector space (TVS) X that is a locally convex inductive limit of a countable inductive system of Fréchet spaces. This means that X is a direct limit of a direct system in the category of locally convex topological vector spaces and each is a Fréchet space. The name LF stands for Limit of Fréchet spaces.
In mathematics, nuclear spaces are topological vector spaces that can be viewed as a generalization of finite-dimensional Euclidean spaces and share many of their desirable properties. Nuclear spaces are however quite different from Hilbert spaces, another generalization of finite-dimensional Euclidean spaces. They were introduced by Alexander Grothendieck.
In functional analysis and related areas of mathematics, a complete topological vector space is a topological vector space (TVS) with the property that whenever points get progressively closer to each other, then there exists some point towards which they all get closer. The notion of "points that get progressively closer" is made rigorous by Cauchy nets or Cauchy filters, which are generalizations of Cauchy sequences, while "point towards which they all get closer" means that this Cauchy net or filter converges to The notion of completeness for TVSs uses the theory of uniform spaces as a framework to generalize the notion of completeness for metric spaces. But unlike metric-completeness, TVS-completeness does not depend on any metric and is defined for all TVSs, including those that are not metrizable or Hausdorff.
The theorem on the surjection of Fréchet spaces is an important theorem, due to Stefan Banach, that characterizes when a continuous linear operator between Fréchet spaces is surjective.
In mathematics, the injective tensor product of two topological vector spaces (TVSs) was introduced by Alexander Grothendieck and was used by him to define nuclear spaces. An injective tensor product is in general not necessarily complete, so its completion is called the completed injective tensor products. Injective tensor products have applications outside of nuclear spaces. In particular, as described below, up to TVS-isomorphism, many TVSs that are defined for real or complex valued functions, for instance, the Schwartz space or the space of continuously differentiable functions, can be immediately extended to functions valued in a Hausdorff locally convex TVS without any need to extend definitions from real/complex-valued functions to -valued functions.
In functional analysis, a branch of mathematics, two methods of constructing normed spaces from disks were systematically employed by Alexander Grothendieck to define nuclear operators and nuclear spaces. One method is used if the disk is bounded: in this case, the auxiliary normed space is with norm The other method is used if the disk is absorbing: in this case, the auxiliary normed space is the quotient space If the disk is both bounded and absorbing then the two auxiliary normed spaces are canonically isomorphic.
In the mathematical discipline of functional analysis, a differentiable vector-valued function from Euclidean space is a differentiable function valued in a topological vector space (TVS) whose domains is a subset of some finite-dimensional Euclidean space. It is possible to generalize the notion of derivative to functions whose domain and codomain are subsets of arbitrary topological vector spaces (TVSs) in multiple ways. But when the domain of a TVS-valued function is a subset of a finite-dimensional Euclidean space then many of these notions become logically equivalent resulting in a much more limited number of generalizations of the derivative and additionally, differentiability is also more well-behaved compared to the general case. This article presents the theory of -times continuously differentiable functions on an open subset of Euclidean space , which is an important special case of differentiation between arbitrary TVSs. This importance stems partially from the fact that every finite-dimensional vector subspace of a Hausdorff topological vector space is TVS isomorphic to Euclidean space so that, for example, this special case can be applied to any function whose domain is an arbitrary Hausdorff TVS by restricting it to finite-dimensional vector subspaces.
In functional analysis and related areas of mathematics, the strong dual space of a topological vector space (TVS) is the continuous dual space of equipped with the strong (dual) topology or the topology of uniform convergence on bounded subsets of where this topology is denoted by or The coarsest polar topology is called weak topology. The strong dual space plays such an important role in modern functional analysis, that the continuous dual space is usually assumed to have the strong dual topology unless indicated otherwise. To emphasize that the continuous dual space, has the strong dual topology, or may be written.
In functional analysis, a topological homomorphism or simply homomorphism is the analog of homomorphisms for the category of topological vector spaces (TVSs). This concept is of considerable importance in functional analysis and the famous open mapping theorem gives a sufficient condition for a continuous linear map between Fréchet spaces to be a topological homomorphism.
In functional analysis and related areas of mathematics, a metrizable topological vector space (TVS) is a TVS whose topology is induced by a metric. An LM-space is an inductive limit of a sequence of locally convex metrizable TVS.