Locally convex topological vector space

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.
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, 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 is a vector space over formula_1 a subfield of the complex numbers (normally formula_2 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 in is called
A topological vector space is called locally convex if the origin has a neighborhood basis (i.e. a local base) consisting of convex sets.
In fact, every locally convex TVS has a neighborhood basis of the origin consisting of sets (i.e. disks), 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 convex. It is possible for a TVS to have neighborhoods of the origin that are convex and yet 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 is a map formula_12 such that
If satisfies positive definiteness, which states that if formula_18 then formula_19, 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 formula_1 where formula_30 is either the real or complex numbers, and let formula_31 (resp. formula_32 denote the open (resp. closed) ball of radius formula_8 in formula_34 
A family of seminorms formula_20 on the vector space induces a canonical vector space topology on , called the initial topology induced by the seminorms, making it into a topological vector space (TVS). By definition, it is the coarsest topology on for which all maps in formula_20 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 definition given above because each formula_37 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 to be induced by a family of norms but for to be normable (that is, to have its topology be induced by a single norm).
Basis and subbasis.
Suppose that formula_20 is a family of seminorms on that induces a locally convex topology 𝜏 on . 
A subbasis at the origin is given by all sets of the form formula_39 as ranges over formula_20 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 formula_41 is a neighborhood of 0 if and only if formula_42 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 formula_20 and every formula_44 let
Bases of seminorms and saturated families.
If is a locally convex space and if formula_20 is a collection of continuous seminorms on , then formula_20 is called a base of continuous seminorms if it is a base of seminorms for the collection of continuous seminorms on . Explicitly, this means that for all continuous seminorms on , there exists a formula_48 and a real formula_8 such that formula_27
If formula_20 is a base of continuous seminorms for a locally convex TVS then the family of all sets of the form formula_52 as varies over formula_20 and varies over the positive real numbers, is a of neighborhoods of the origin in (not just a subbasis, so there is no need to take finite intersections of such sets).
A family formula_20 of seminorms on a vector space is called saturated if for any and in formula_20, the seminorm defined by formula_56 belongs to formula_20.
If formula_20 is a saturated family of continuous seminorms that induces the topology on then the collection of all sets of the form formula_59 as ranges over formula_20 and ranges over all positive real numbers, forms a neighborhood basis at the origin consisting of convex open sets; 
This forms a basis at the origin rather than merely a subbasis so that in particular, there is need to take finite intersections of such sets.
Basis of norms.
The following theorem implies that if is a locally convex space then the topology of can be a defined by a family of continuous on (a norm is a seminorm formula_61 where formula_62 implies formula_63) if and only if there exists continuous on . 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 formula_20 of seminorms (each is which is necessarily continuous) then the family formula_65 of (also continuous) norms obtained by adding some given continuous norm formula_66 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 then is necessarily Hausdorff but the converse is not in general true (not even for locally convex spaces or Fréchet spaces).
Nets.
Suppose that the topology of a locally convex space is induced by a family formula_20 of continuous seminorms on . 
If formula_68 and if formula_69 is a net in , then formula_70 in if and only if for all formula_71 formula_72 
Moreover, if formula_73 is Cauchy in , then so is formula_74 for every formula_75
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 formula_3, define the Minkowski functional of to be
From this definition it follows that formula_78 is a seminorm if 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.
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 (HBEP) 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, formula_20 is a family of continuous seminorms that generate the topology of .
Properties of convex hulls.
For any subset of a TVS , the convex hull (resp. closed convex hull, balanced hull, resp. convex balanced hull) of , denoted by formula_106 (resp. formula_107 formula_108, formula_109), is the smallest convex (resp. closed convex, balanced, convex balanced) subset of containing .
Examples and nonexamples.
Finest and coarsest locally convex topology.
Coarsest vector topology.
Any vector space 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 formula_135 
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 formula_135
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 formula_20 is the set of all seminorms on then the locally convex TVS topology, denoted by , that formula_20 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 . 
 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 (and thus not metrizable). 
Moreover, is the Hausdorff locally convex topology on with the property that any linear map from it into any Hausdorff locally convex space is continuous.
The space is a bornological space.
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 formula_139 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 formula_141, embedded in formula_139 in the natural way, by completing finite sequences with infinitely many formula_143.
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 formula_144 for all in .
Spaces of differentiable functions give other non-normable examples. 
Consider the space of smooth functions formula_145 such that formula_146, where "a" and "b" are multiindices. 
The family of seminorms defined by formula_147 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 formula_148 
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 formula_149 of functions with is a Fréchet space with countable family of seminorms (these are actually norms, and the completion of the space formula_149 with the norm is a Banach space ). 
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 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 formula_151 is the space of distributions on formula_152
More abstractly, given a topological space , the space formula_153 of continuous (not necessarily bounded) functions on can be given the topology of uniform convergence on compact sets. 
This topology is defined by semi-norms (as varies over the directed set of all compact subsets of ). 
When is locally compact (e.g. an open set in formula_141) the Stone-Weierstrass theorem applies—in the case of real-valued functions, any subalgebra of formula_153 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 . In particular, their dual space is trivial, that is, it contains only the zero functional.
Continuous 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 and with families of seminorms and respectively, a linear map formula_163 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.
If is a real or complex vector space, is a linear functional on , and is a seminorm on , then formula_166 if and only if formula_167
If is a non-0 linear functional on a real vector space and if is a seminorm on , then formula_168 if and only if formula_169
Multilinear maps.
Let formula_170 be an integer, formula_171 be TVSs (not necessarily locally convex), let be a locally convex TVS whose topology is determined by a family formula_21 of continuous seminorms, and let formula_173 be a multilinear operator that is linear in each of its coordinates. 
The following are equivalent: 