In functional analysis and related areas of mathematics, locally convex topological vector spaces 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
- History
- Definition
- Definition via convex sets
- Definition via seminorms
- Equivalence of definitions
- Examples of locally convex spaces
- Examples of spaces lacking local convexity
- Continuous linear mappings
- References
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 Frechet'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 V is a vector space over K, a subfield of the complex numbers (normally C 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 subset C in V is called
- Convex if for all x, y in C, and 0 ≤ t ≤ 1, tx + (1 – t)y is in C. In other words, C contains all line segments between points in C.
- Circled if for all x in C, λx is in C if | λ | = 1. If K = R, this means that C is equal to its reflection through the origin. For K = C, 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.
- A cone (when the underlying field is ordered) if for all x in C and 0 ≤ λ ≤ 1, λx is in C.
- Balanced if for all x in C, λx is in C if | λ | ≤ 1. If K = R, this means that if x is in C, C contains the line segment between x and −x. For K = C, 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.
- Absorbent or absorbing if the union of tC over all t > 0 is all of V, or equivalently for every x in V, tx is in C for some t > 0. The set C can be scaled out to absorb every point in the space.
- Absolutely convex if it is both balanced and convex.
More succinctly, a subset of V is absolutely convex if it is closed under linear combinations whose coefficients absolutely sum to ≤ 1. Such a set is absorbent if it spans all of V.
Definition (first version). A topological vector space is called locally convex if the origin has a local base of absolutely convex absorbent sets.
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 V is a map p : V → R such that
- p is positive or positive semidefinite: p(x) ≥ 0.
- p is positive homogeneous or positive scalable: p(λx) = | λ | p(x) for every scalar λ. So, in particular, p(0) = 0.
- p is subadditive. It satisfies the triangle inequality: p(x + y) ≤ p(x) + p(y).
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.
Definition (second version). A locally convex space is defined to be a vector space V along with a family of seminorms {pα}α ∈ A on V.
A locally convex space carries a natural topology, called the initial topology induced by the seminorms. By definition, it is the coarsest topology for which all the mappings
are continuous. A base of neighborhoods of y for this topology is obtained in the following way: for every finite subset B of A and every ε > 0, let
Note that
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 UB, ε (0) is absolutely convex and absorbent (and because the latter properties are preserved by translations).
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 0 ≤ t ≤ 1, define the Minkowski functional of C to be
From this definition it follows that μC 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.
Examples of locally convex spaces
0(U) is not complete in the uniform norm. The topology on D(U) is defined as follows: for any fixed compact set K ⊂ U, the space C∞
0(K) of functions f ∈ C∞
0(U) with supp( f ) ⊂ K is a Fréchet space with countable family of seminorms || f ||m = supx| Dm f (x) | (these are actually norms, and the space C∞
0(K) 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 D(Rn) is the space of distributions on Rn.
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 linear 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 V and W with families of seminorms {pα}α and {qβ}β respectively, a linear map T : V → W is continuous if and only if for every β, there exist α1, α2, ..., αn and M > 0 such that for all v in V
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.