Locally convex topological vector space

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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.

Contents

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.

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, [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).

Definition

Suppose X 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 subset C in X is called

  1. Convex if for all x, y in C, and tx + (1 – t)y is in C. In other words, C contains all line segments between points in C.
  2. Circled if for all x in C, λx is in C if |λ| = 1. If this means that C is equal to its reflection through the origin. For it means for any x in C, C contains the circle through x, centred on the origin, in the one-dimensional complex subspace generated by x.
  3. A cone (when the underlying field is ordered) if for all x in C and 0 ≤ λ ≤ 1,λx is in C.
  4. Balanced if for all x in C, λx is in C if |λ| ≤ 1. If this means that if x is in C, C contains the line segment between x and x. For it means for any x in C, C contains the disk with x on its boundary, centred on the origin, in the one-dimensional complex subspace generated by x. Equivalently, a balanced set is a circled cone.
  5. Absorbent or absorbing if for every x in X, there exists such that x is in tC for all satisfying The set C can be scaled out by any "large" value to absorb every point in the space.
    • In any TVS, every neighborhood of the origin is absorbent. [7]
  6. Absolutely convex or a disk if it is both balanced and convex. This is equivalent to it being closed under linear combinations whose coefficients absolutely sum to ; such a set is absorbent if it spans all of X.

A topological vector space is called locally convex if the origin has a neighborhood basis (i.e. a local base) consisting of convex sets. [7]

In fact, every locally convex TVS has a neighborhood basis of the origin consisting of absolutely convex sets (i.e. disks), where this neighborhood basis can further be chosen to also consist entirely of open sets or entirely of closed sets. [7] 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 translation is (by definition of "topological vector space") continuous, 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

  1. p is positive or positive semidefinite: ;
  2. p is positive homogeneous or positive scalable: for every scalar So, in particular, ;
  3. p is subadditive. It satisfies the triangle inequality:

If p satisfies positive definiteness, which states that if then , 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.

Definition: If X is a vector space and is a family of seminorms on X then a subset of is called a base of seminorms for if for all there exists a and a real such that [8]
Definition (second version): A locally convex space is defined to be a vector space X along with a family of seminorms on X.

Seminorm topology

Suppose that X is a vector space over where is either the real or complex numbers, and let (resp. denote the open (resp. closed) ball of radius in A family of seminorms 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 are continuous.

That the vector space operations are continuous in this topology follows from properties 2 and 3 above. It can easily be seen that the resulting topological vector space is "locally convex" in the sense of the first definition given above because each is absolutely convex and absorbent (and because the latter properties are preserved by translations).

It is possible for a locally convex topology on a space X to be induced by a family of norms but for X to not be normable (that is, to have its topology be induced by a single norm).

Basis and subbasis

Suppose that is a family of seminorms on X that induces a locally convex topology 𝜏 on X. A subbasis at the origin is given by all sets of the form as p ranges over and r ranges over the positive real numbers. A base at the origin is given by the collection of all possible finite intersections of such subbasis sets.

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 S is a neighborhood of 0 if and only if 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 and every let

Bases of seminorms and saturated families

If X is a locally convex space and if is a collection of continuous seminorms on X, then is called a base of continuous seminorms if it is a base of seminorms for the collection of all continuous seminorms on X. [8] Explicitly, this means that for all continuous seminorms p on X, there exists a and a real such that [8]

If 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 and r varies over the positive real numbers, is a base of neighborhoods of the origin in X (not just a subbasis, so there is no need to take finite intersections of such sets). [8]

A family of seminorms on a vector space X is called saturated if for any p and q in , the seminorm defined by belongs to .

If 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 and r ranges over all positive real numbers, forms a neighborhood basis at the origin consisting of convex open sets; [8] 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. [8]

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 norms on X (a norm is a seminorm where implies ) if and only if there exists at least one continuous norm on X. [9] 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 is 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 X then X is necessarily Hausdorff but the converse is not in general true (not even for locally convex spaces or Fréchet spaces).

Theorem [10]   Let be a Fréchet space over the field Then the following are equivalent:

  1. does not admit a continuous norm (that is, any continuous seminorm on can not be a norm).
  2. contains a vector subspace that is TVS-isomorphic to
  3. contains a complemented vector subspace that is TVS-isomorphic to
Nets

Suppose that the topology of a locally convex space X is induced by a family of continuous seminorms on X. If and if is a net in X, then in X if and only if for all [11] Moreover, if is Cauchy in X, then so is for every [11]

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 ε-balls is the triangle inequality.

For an absorbing set C such that if x is in C, then tx is in C whenever , define the Minkowski functional of C to be

From this definition it follows that 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

Theorem [7]   Suppose that X is a (real or complex) vector space and let be a filter base of subsets of X such that:

  1. Every is convex, balanced, and absorbing;
  2. For every there exists some real r satisfying such that

Then is a neighborhood base at 0 for a locally convex TVS topology on X.

Theorem [7]   Suppose that X is a (real or complex) vector space and let be a non-empty collection of convex, balanced, and absorbing subsets of X. Then the set of all of all positive scalar multiples of finite intersections of sets in forms a neighborhood base at 0 for a locally convex TVS topology on X.

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. [12] Say that X has the Hahn-Banach extension property (HBEP) if every vector subspace of X has the extension property. [12]

The Hahn-Banach theorem guarantees that every Hausdorff locally convex space has the HBEP. For complete metrizable TVSs there is a converse:

Theorem [12]  (Kalton)  Every complete metrizable TVS with the Hahn-Banach extension property is locally convex.

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. [12]

Properties

Throughout, is a family of continuous seminorms that generate the topology of X.

Topological properties

Topological properties of convex subsets

Properties of convex hulls

For any subset S of a TVS X, the convex hull (resp. closed convex hull, balanced hull , resp. convex balanced hull) of S, denoted by (resp. , ), is the smallest convex (resp. 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 (i.e. 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 X if and only 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 is the set of all seminorms on X then the locally convex TVS topology, denoted by 𝜏lc, that induces on X is called the finest locally convex topology on X. [28] This topology may also be described as the TVS-topology on X having as a neighborhood base at 0 the set of all absorbing disks in X. [28] Any locally convex TVS-topology on X is necessarily a subset of 𝜏lc. (X, 𝜏lc) is Hausdorff. [15] Every linear map from (X, 𝜏lc) into another locally convex TVS is necessarily continuous. [15] In particular, every linear functional on (X, 𝜏lc) is continuous and every vector subspace of X is closed in (X, 𝜏lc).; [15] therefore, if X is infinite dimensional then (X, 𝜏lc) is not pseudometrizable (and thus not metrizable). [28] Moreover, 𝜏lc is the only Hausdorff locally convex topology on X with the property that any linear map from it into any Hausdorff locally convex space is continuous. [29] The space (X, 𝜏lc) is a bornological space. [30]

Examples of locally convex spaces

Every normed space is a Hausdorff locally convex space, and much of the theory of locally convex spaces generalises 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 Lp spaces with p ≥ 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 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 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 metrisable 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 C
0
(U)
is not complete in the uniform norm. The topology on D(U) is defined as follows: for any fixed compact set KU, the space of functions fC
0
(U)
with supp(f) ⊂ K is a Fréchet space with countable family of seminorms ||f||m = supk≤msupx|Dkf(x)| (these are actually norms, and the completion of the space with the ||||m norm is a Banach space Dm(K)). Given any collection {Kλ}λ of compact sets, directed by inclusion and such that their union equal U, the C
0
(Kλ)
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 C
0
(Kλ)
with the strongest locally convex topology which makes each inclusion map C
0
(Kλ) ↪ D(U)
continuous. This space is locally convex and complete. However, it is not metrisable, and so it is not a Fréchet space. The dual space of is the space of distributions on

More abstractly, given a topological space X, the space 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 φK(f) = max{|f(x)| : xK }(as K varies over the directed set of all compact subsets of X). When X is locally compact (e.g. 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 (e.g., 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

Theorem [31]   Let be a linear operator between TVSs where Y is locally convex (note that X need not be locally convex). Then is continuous if and only if for every continuous seminorm q on Y, there exists a continuous seminorm p on X 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 X and Y with families of seminorms {pα}α and {qβ}β respectively, a linear map is continuous if and only if for every β, there exist α1, α2, …, αn 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 {pα}α 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

Theorem [31]   If X is a TVS (not necessarily locally convex) and if f is a linear functional on X, then f is continuous if and only if there exists a continuous seminorm p on X such that

If X is a real or complex vector space, f is a linear functional on X, and p is a seminorm on X, then if and only if [31] If f is a non-0 linear functional on a real vector space X and if p is a seminorm on X, then if and only if [15]

Multilinear maps

Let be an integer, be TVSs (not necessarily locally convex), let Y 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 n coordinates. The following are equivalent:

  1. M is continuous.
  2. For every , there exist continuous seminorms on respectively, such that for all [15]
  3. For every , there exists some neighborhood of 0 in on which is bounded. [15]

See also

Notes

    1. Hausdorff, F. Grundzüge der Mengenlehre (1914)
    2. von Neumann, J. Collected works. Vol II. p.94-104
    3. Dieudonne, J. History of Functional Analysis Chapter VIII. Section 1.
    4. von Neumann, J. Collected works. Vol II. p.508-527
    5. Dieudonne, J. History of Functional Analysis Chapter VIII. Section 2.
    6. Banach, S. Theory of linear operations p.75. Ch. VIII. Sec. 3. Theorem 4., translated from Theorie des operations lineaires (1932)
    7. 1 2 3 4 5 6 7 8 9 Narici & Beckenstein 2011, pp. 67-113.
    8. 1 2 3 4 5 6 Narici & Beckenstein 2011, p. 122.
    9. Jarchow 1981, p. 130.
    10. Jarchow 1981, pp. 129-130.
    11. 1 2 Narici & Beckenstein 2011, p. 126.
    12. 1 2 3 4 Narici & Beckenstein 2011, pp. 225-273.
    13. Narici & Beckenstein 2011, pp. 177-220.
    14. Narici & Beckenstein 2011, p. 149.
    15. 1 2 3 4 5 6 7 Narici & Beckenstein 2011, pp. 149-153.
    16. Narici & Beckenstein 2011, pp. 115-154.
    17. 1 2 3 4 5 6 Trèves 2006, p. 126.
    18. 1 2 Schaefer & Wolff 1999, p. 38.
    19. Conway 1990, p. 102.
    20. Trèves 2006, p. 370.
    21. 1 2 Narici & Beckenstein 2011, pp. 155-176.
    22. Rudin 1991, p. 7.
    23. Trèves 2006, p. 67.
    24. Trèves 2006, p. 145.
    25. Trèves 2006, p. 362.
    26. 1 2 Trèves 2006, p. 68.
    27. 1 2 Dunford 1988, p. 415.
    28. 1 2 3 Narici & Beckenstein 2011, pp. 125-126.
    29. Narici & Beckenstein 2011, p. 476.
    30. Narici & Beckenstein 2011, p. 446.
    31. 1 2 3 Narici & Beckenstein 2011, pp. 126-128.

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