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 (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 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 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 Template:Mvar 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.

Convex sets

A subset Template:Mvar in Template:Mvar is called

1. Convex if for all x, y in Template:Mvar, and 0 ≤ t ≤ 1, tx + (1 – t)y is in Template:Mvar. In other words, Template:Mvar contains all line segments between points in Template:Mvar.
2. Circled if for all Template:Mvar in Template:Mvar, λx is in Template:Mvar if |λ| = 1. If K = R, this means that Template:Mvar is equal to its reflection through the origin. For K = C, it means for any Template:Mvar in Template:Mvar, Template:Mvar contains the circle through Template:Mvar, centred on the origin, in the one-dimensional complex subspace generated by Template:Mvar.
3. A cone (when the underlying field is ordered) if for all Template:Mvar in Template:Mvar and 0 ≤ λ ≤ 1, λx is in Template:Mvar.
4. Balanced if for all Template:Mvar in Template:Mvar, λx is in Template:Mvar if |λ| ≤ 1. If K = R, this means that if Template:Mvar is in Template:Mvar, Template:Mvar contains the line segment between Template:Mvar and −x. For K = C, it means for any Template:Mvar in Template:Mvar, Template:Mvar contains the disk with Template:Mvar on its boundary, centred on the origin, in the one-dimensional complex subspace generated by Template:Mvar. Equivalently, a balanced set is a circled cone.
5. Absorbent or absorbing if the union of tC over all t > 0 is all of Template:Mvar, or equivalently for every Template:Mvar in Template:Mvar, tx is in Template:Mvar for some t > 0. The set Template:Mvar can be scaled out to absorb every point in the space.
6. Absolutely convex if it is both balanced and convex.

More succinctly, a subset of Template:Mvar 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 Template:Mvar.

A locally convex topological vector space is a topological vector space in which 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.

Seminorms

A seminorm on Template:Mvar is a map p : V → R such that

1. Template:Mvar is positive or positive semidefinite: p(x) ≥ 0.
2. Template:Mvar is positive homogeneous or positive scalable: p(λx) = |λ| p(x) for every scalar Template:Mvar. So, in particular, p(0) = 0.
3. Template:Mvar is subadditive. It satisfies the triangle inequality: p(x + y) ≤ p(x) + p(y).

If Template:Mvar satisfies positive definiteness, which states that if p(x) = 0 then x = 0, then Template:Mvar 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.

A locally convex space is then defined to be a vector space Template:Mvar along with a family of seminorms Template:Mset_{α ∈ A} on Template:Mvar. The space carries a natural topology, the initial topology of the seminorms. In other words, it is the coarsest topology for which all the mappings

${\begin{cases}p_{\alpha ,y}:V\to \mathbf {R} \\x\mapsto p_{\alpha }(x-y)&y\in V,\alpha \in A\end{cases}}$

are continuous. A base of neighborhoods of Template:Mvar for this topology is obtained in the following way: for every finite subset Template:Mvar of Template:Mvar and every ε > 0, let

$U_{B,\varepsilon }(y)=\{x\in V:p_{\alpha }(x-y)<\varepsilon \ \forall \alpha \in B\}.$

Note that

$U_{B,\varepsilon }(y)=\bigcap _{\alpha \in B}(p_{\alpha ,y})^{-1}([0,\varepsilon )).$

That the vector space operations are continuous in this topology follows from properties 2 and 3 above. The resulting TVS is locally convex because each U_B, 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 Template:Mvar-balls is the triangle inequality.

For an absorbing set Template:Mvar such that if Template:Mvar is in Template:Mvar, then tx is in Template:Mvar whenever 0 ≤ t ≤ 1, define the Minkowski functional of Template:Mvar to be

$\mu _{C}(x)=\inf\{\lambda >0:x\in \lambda C\}.$

From this definition it follows that μ_C is a seminorm if Template:Mvar is balanced and convex (it is also absorbent by assumption). Conversely, given a family of seminorms, the sets

$\left\{x:p_{\alpha _{1}}(x)<\varepsilon ,\cdots ,p_{\alpha _{n}}(x)<\varepsilon \right\}$

form a base of convex absorbent balanced sets.

Further definitions and properties

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Examples and nonexamples

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 L^p 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 R^ω of real valued sequences with the family of seminorms given by

$p_{i}\left(\left\{x_{n}\right\}_{n}\right)=\left|x_{i}\right|,\qquad i\in \mathbf {N} .$

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 Rⁿ, embedded in R^ω in the natural way, by completing finite sequences with infinitely many 0.

• Given any vector space Template:Mvar and a collection Template:Mvar of linear functionals on it, Template:Mvar can be made into a locally convex topological vector space by giving it the weakest topology making all linear functionals in Template:Mvar continuous. This is known as the weak topology or the initial topology determined by Template:Mvar. The collection Template:Mvar may be the algebraic dual of Template:Mvar or any other collection. The family of seminorms in this case is given by p_f (x) = | f (x)| for all Template:Mvar in Template:Mvar.

• Spaces of differentiable functions give other non-normable examples. Consider the space of smooth functions f : Rⁿ → C such that sup_x|x^aD^bf | < ∞, where a and b are multiindices. The family of seminorms defined by p_a,b( f ) = sup_x |x^aD^bf(x)| 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 U ⊆ Rⁿ. A more detailed construction is needed for the topology of this space because the space CTemplate:Su(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 CTemplate:Su(K) of functions f ∈ CTemplate:Su(U) with supp( f ) ⊂ K is a Fréchet space with countable family of seminorms Template:Norm_m = sup_x|D^m f (x)| (these are actually norms, and the space CTemplate:Su(K) with the Template:Norm_m norm is a Banach space D^m(K)). Given any collection Template:Mset_λ of compact sets, directed by inclusion and such that their union equal Template:Mvar, the CTemplate:Su(K_λ) 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 CTemplate:Su(K_λ) with the final topology which makes each inclusion map CTemplate:Su(K_λ) ↪ 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.

• More abstractly, given a topological space Template:Mvar, the space C(X) of continuous (not necessarily bounded) functions on Template:Mvar can be given the topology of uniform convergence on compact sets. This topology is defined by semi-norms φ_K( f ) = maxTemplate:Mset (as Template:Mvar varies over the directed set of all compact subsets of Template:Mvar). When Template:Mvar is locally compact (e.g. an open set in Rⁿ) 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 (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:

• The spaces L^p([0, 1]) for 0 < p < 1 are equipped with the F-norm

$\|f\|_{p}=\int _{0}^{1}|f(x)|^{p}\,dx$

they are not locally convex, since the only convex neighborhood of zero is the whole space. More generally the spaces L^p(μ) with an atomless, finite measure Template:Mvar and 0 < p < 1 are not locally convex.

• The space of measurable functions on the unit interval [0, 1] (where we identify two functions that are equal almost everywhere) has a vector-space topology defined by the translation-invariant metric: (which induces the convergence in measure of measurable functions; for random variables, convergence in measure is convergence in probability)

$d(f,g)=\int _{0}^{1}{\frac {|f(x)-g(x)|}{1+|f(x)-g(x)|}}\,dx.$

This space is often denoted L₀.

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.

• The sequence space ℓ^p, 0 < p < 1, is not locally convex.

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 Template:Mvar and Template:Mvar with families of seminorms Template:Mset_α and Template:Mset_β respectively, a linear map T : V → W is continuous if and only if for every Template:Mvar, there exist α₁, α₂, ..., α_n and M > 0 such that for all Template:Mvar in Template:Mvar

$q_{\beta }(Tv)\leq M\left(p_{\alpha _{1}}(v)+\dotsb +p_{\alpha _{n}}(v)\right).$

In other words, each seminorm of the range of Template:Mvar is bounded above by some finite sum of seminorms in the domain. If the family Template:Mset_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:

$q_{\beta }(Tv)\leq Mp_{\alpha }(v).$

The class of all locally convex topological vector spaces forms a category with continuous linear maps as morphisms.

See also

• Krein–Milman theorem

References

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