Locally convex topological vector space


In functional analysis and related areas of mathematics, locally convex topological vector spaces or locally convex spaces are examples of topological vector spaces 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. 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.
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.
A notable example of a result which had to wait for the development and dissemination of general locally convex spaces 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.

Definition

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

Definition via convex sets

A subset in is called
  1. Convex if for all in, and is in. In other words, contains all line segments between points in.
  2. Circled if for all in, is in if. If, this means that is equal to its reflection through the origin. For, it means for any in, contains the circle through, centred on the origin, in the one-dimensional complex subspace generated by.
  3. A cone if for all in and is in.
  4. Balanced if for all in, is in if. If, this means that if is in, contains the line segment between and. For, it means for any in, contains the disk with on its boundary, centred on the origin, in the one-dimensional complex subspace generated by. Equivalently, a balanced set is a circled cone.
  5. Absorbent or absorbing if for every in, there exists such that is in for all satisfying. The set can be scaled out by any "large" value to absorb every point in the space.
  6. * In any TVS, every neighborhood of the origin is absorbent.
  7. 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.
Definition: A topological vector space is called locally convex if the origin has a neighborhood basis consisting of convex sets.
In fact, every locally convex TVS has a neighborhood basis of the origin consisting of absolutely convex sets, 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 and convex.
Note that 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 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 is a map such that

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

Seminorm topology

Suppose that is a vector space over, where is either the real or complex numbers, and let denote the open ball of radius in.
A family of seminorms on vector space induces a canonical vector space topology on, called the initial topology induced by the seminorms, making it into a topological vector space.
By definition, it is the coarsest topology on 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.
Note that 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.
Basis and subbasis
Suppose that is a family of seminorms on that induces a locally convex topology ? on.
A subbasis at the origin is given by all sets of the form as ranges over and 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 is any subset of containing the origin then for any, is a neighborhood of 0 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 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 .
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;
note that 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.
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 is in, then is in whenever, define the Minkowski functional of to be
From this definition it follows that is a seminorm if is balanced and convex. Conversely, given a family of seminorms, the sets
form a base of convex absorbent balanced sets.

Ways of defining a locally convex topology

Further definitions

Sufficient conditions

;Hahn-Banach extension property
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.
Say that has the Hahn-Banach extension property if every vector subspace of 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 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, is a family of continuous seminorms that generate the topology of.

Topological properties

Topological properties of convex subsets

Properties of convex hulls

For any subset of a TVS, the convex hull of, denoted by , is the smallest convex subset of containing.

Examples and nonexamples

Finest and coarsest locally convex topology

;Coarsest vector topology
Any vector space endowed with the trivial topology is a locally convex TVS.
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.
;Finest locally convex topology
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.
This topology may also be described as the TVS-topology on having as a neighborhood base at 0 the set of all absorbing disks in.
Any locally convex TVS-topology on is necessarily a subset of ?lc.
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 is closed in.;
therefore, if is infinite dimensional then is not pseudometrizable.
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.

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 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. Note that 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 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 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 .
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 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, the space of continuous functions on can be given the topology of uniform convergence on compact sets.
This topology is defined by semi-norms.
When is locally compact the Stone-Weierstrass theorem applies—in the case of real-valued functions, any subalgebra of that separates points and contains the constant functions 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. In particular, their dual space is trivial, that is, it contains only the zero functional.
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
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.

Linear functionals

Note that if is a real or complex vector space, is a linear functional on, and is a seminorm on, then if and only if.
If is a non-0 linear functional on a real vector space and if is a seminorm on, then if and only if.

Multilinear maps

Let be an integer, be TVSs, 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:

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