LOCALLY CONVEX TOPOLOGICAL VECTOR SPACE

(Redirected from Locally convex 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) which generalise normed spaces and metric vector spaces. They can be defined as topological vector spaces which have a base 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 which are metrisable and complete with respect to this metric. They are generalisations of Banach spaces, which are complete vector spaces with respect to a norm.

Contents

Definition

Convex sets

Seminorms

Equivalence of definitions

Further definitions and properties

Examples and nonexamples

Examples of locally convex spaces

Nonexamples of locally convex spaces

Continuous linear mappings

See also

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.

Convex sets

A subset ''C'' in ''V'' is called
# Convex if for each ''x'' and ''y'' in ''C'', ''tx''+(1–''t'')''y'' is in ''C'' for all ''t'' in the unit interval, that is whenever 0 ≤ ''t'' ≤ 1. In other words, ''C'' contains all line segments between points in ''C''.
# Circled if for all ''x'' in ''C'', λ''x'' is in ''C'' if |λ|=1. If the underlying field 'K' is the real numbers, this means that ''C'' is equal to its reflection through the origin. For a complex vector space ''V'', 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 every ''x'' in ''C'' and 0 ≤ λ ≤ 1, λ''x'' is in ''C''.
# Balanced if for all ''x'' in ''C'', λ''x'' is in ''C'' if |λ| ≤ 1. If the underlying field 'K' is the real numbers, this means that if ''x'' is in ''C'', ''C'' contains the line segment between ''x'' and ''-x''. For a complex vector space ''V'', 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.
A locally convex topological vector space is a topological vector space with a base of ''absolutely convex absorbent'' sets. Because it is a vector space, every local base gives rise to a base by translation, so it is sufficient to supply a local base of the origin. Therefore a locally convex topological vector space is one in which every open neighborhood of the zero vector contains a balanced absorbent convex set as a subset.

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''(λ''x'') = |λ|''p''(''x'')
# ''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.
A locally convex space is then defined to be a vector space ''V'' along with a family of seminorms {''p''_α}_α on ''V''. The space carries a natural topology, the initial topology of the seminorms and the vector space operations of addition and scalar multiplication. In other words, it is the coarsest topology for which all the seminorms and the vector space operations are continuous.

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 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
:$mu\_C(x)\; =\; inf\; \{lambda\; >\; 0:\; xisin\; lambda\; C\}.$
From this definition it follows that μ_''C'' 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.

Further definitions and properties

• A family of seminorms {''p''_α}_α is called 'total' or 'separated' or is said to 'separate points' if whenever ''p''_α(''x'') = 0 holds for every α then ''x'' is necessarily 0. A locally convex space is Hausdorff if and only if it has a separated family of seminorms. Many authors take the Hausdorff criterion in the definition.


• A pseudometric is a generalisation of a metric which does not satisfy the identity of indiscernables condition which requires that ''d''(''x'',''y'') = 0 only when ''x''=''y''. A locally convex space is pseudometrisable, meaning that its topology arises from a pseudometric, if and only if it has a countable family of seminorms. Indeed, a pseudometric inducing the same topology is then given by
  ::
  (where the 1/2ⁿ can be replaced by any positive summable sequence ''a_n'').
  This pseudometric is translation-invariant, but not homogeneous, meaning ''d''(''kx'',''ky'') does not equal |''k''|''d''(''x'',''y''), and therefore does not
  define a (pseudo)norm.
  The pseudometric is an honest metric if and only if the family of seminorms is separated,
  since this is the case if and only if the space is Hausdorff.
  If furthermore the space is complete, the space is called a Fréchet space.
  


• As with any topological vector space, a locally convex space is also a uniform space. Thus one may speak of uniform continuity, uniform convergence, and Cauchy sequences.


• A Cauchy net (or filter) in a locally convex space is a net {''x''_κ}_κ such that for every ε > 0 and every seminorm ''p''_α, there exists a κ such that for every λ, μ > κ, ''p''_α(''x''_λ–''x''_μ) < ε. In other words, the net must be Cauchy in all the seminorms simultaneously. The definition of completeness is given here in terms of nets instead of the more familiar sequences because unlike Fréchet spaces which are metrisable and therefore first-countable, locally convex spaces will not generally not be first-countable. Sequences, which are countable by definition, cannot suffice to characterize convergence in such spaces. A locally convex space is complete if and only if every Cauchy net converges.


• A family of seminorms becomes a preordered set under the relation ''p''_α ≤ ''p''_β if and only if there exists an ''M'' > 0 such that for all ''x'', ''p''_α(''x'') ≤ ''Mp''_β(''x''). One says it is a 'directed family of seminorms' if the family is a directed set with addition as the join, in other words if for every α and β, there is a γ such that ''p''_α + ''p''_β ≤ ''p''_γ. Every family of seminorms has an equivalent directed family, meaning one which defines the same topology. Indeed, given a family {''p''_α}_α∈''I'', let Φ be the set of finite subsets of ''I'', then for every ''F'' in Φ, define ''q''_''F''=∑_α∈''F'' ''p''_α. One may check that {''q''_''F''}_''F''∈Φ is an equivalent directed family.


• If the topology of the space is induced from a single seminorm, then the space 'seminormable'. Any locally convex space with a finite family of seminorms is seminormable. Moreover, if the space is Hausdorff (the family is separated), then the space is normable, with norm given by the sum of the seminorms. In terms of the open sets, a locally convex topological vector space is seminormable if and only if 0 has a bounded neighborhood.

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''( {''x''_''n''}_''n'' )=|''x''_''i'' |. The countable family of seminorms is complete and separable, so this is a Fréchet space. Note that this is also the limit topology of the spaces 'R'^''n''.

★ Given any vector space ''V'' and a collection ''F'' of linear functionals on it, ''V'' 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''. ''F'' may be the algebraic dual of ''V'' or any other collection. The family of seminorms in this case is given by ''p''_''f''(''x'') = |''f''(''x'')| for all ''f'' in ''F''.

★ Spaces of differentiable functions give other non-normable examples. Consider the space of smooth functions 'R'^''n'' → 'C' such that sup_''x'' |''x''^''a''''D''^''b''| < ∞, where ''a'' and ''b'' are multiindices. The family of seminorms define by ''p''_''a'',''b''(''f'') = sup_''x'' |''x''^''a''''D''^''b''''f''(''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 functions with compact support in ''U'' ⊆ 'R'^''n''. 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 ''K'', the space ''C''^∞₀(''K'') of functions with supp(''f'')=''K'' is a Fréchet space with countable family of seminorms ||''f''||_''m''=sup_''x'' |''D''^''m''''f''(''x'')| (these are actually bona fide norms, and the space ''C''^∞₀(''U'') with the ||
★ ||_''m'' norm is a Banach space ''D''^''m''(''K'')) . Given any collection of {''K''_λ}_λ compact sets such that ∪_λ''K''_λ = ''U'', then the ''C''^∞₀(''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 ''C''^∞₀(''K''_λ) with the final topology which makes each inclusion map ''C''^∞₀(''K''_λ)↪''D''(''U'') continuous. This space is locally convex and sequentially complete, but not complete (this means that all Cauchy sequences converge, but not necessarily more general Cauchy nets). The space is not metrisable, and so it is not a Fréchet space. The dual space of ''D''('R'^''n'') is the space of distributions on 'R'^''n''.

★ More abstractly, given a topological space ''X'', the space ''C(X)'' 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 ''φ_K(f)'' = max { |f(x)| : x ∈ K } (as ''K'' varies over the directed set of all compact subsets of ''X''). When ''X'' is locally compact (e.g. an open set in 'R'^''n'') the Stone-Weierstrass theorem applies -- any subalgebra of ''C(X)'' that separates points (e.g. polynomials) is dense.

Nonexamples of locally convex spaces

It is fair to say that almost all vector spaces one meets are locally convex, but here are some examples which are not.

• ''L''^''p'' for 0<''p''<1 are not locally convex, since the ||
  ★ ||_''p'' map does not satisfy the triangle inequality, and is therefore not a norm. These are examples of F-spaces.


• The space of measurable functions on the unit interval $I$ (where we identify two functions that are equal almost everywhere) has a topology defined by the quasinorm
  :{1+|f(x)|}dx
  This space has the curious property that any continuous linear map to the real numbers is 0. In particular it is not locally convex, and its dual space is 0.

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 {''p''_α}_α and {''q''_β}_β respectively, a linear map ''T'' from ''V'' to ''W'' is continuous if and only if for every β there exist α₁, α₂, ... , α_''n'' and exists an ''M''>0 such that for all ''x'' in ''X''
:
In other words, each seminorm of the range of ''T'' is bounded above by some finite sum of seminorms in the codomain. If the family {''p''_α}_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.

See also

★ Krein-Milman theorem