Stereotype space
In functional analysis and related areas of mathematics, stereotype spaces are topological vector spaces defined by a special variant of reflexivity condition. They form a class of spaces with a series of remarkable properties, in particular, this class is very wide (for instance, it contains all Fréchet spaces and thus, all Banach spaces), it consists of spaces satisfying a natural condition of completeness, and it forms a *-autonomous category with the standard analytical tools for constructing new spaces, like taking dual space, space of operators, tensor products, and in addition, immediate subspace, immediate quotient space, products and coproducts, limits and colimits, etc.
Contents
Definition
A stereotype space^{[1]} is a topological vector space over the field of complex numbers^{[2]} such that the natural map into the second dual space
is an isomorphism of topological vector spaces (i.e. a linear and a homeomorphic map). Here the dual space is defined as the space of all linear continuous functionals endowed with the topology of uniform convergence on totally bounded sets in , and the second dual space is the space dual to in the same sense.
The following criterion holds:^{[3]} a topological vector space is stereotype if and only if it is locally convex and satisfies the following two conditions:
- pseudocompleteness: each totally bounded Cauchy net in converges,
- pseudosaturateness: each closed convex balanced capacious^{[4]} set in is a neighborhood of zero in .
The property of being pseudocomplete is a weakening of the usual notion of completeness, while the property of being pseudosaturated is a weakening of the notion of barreledness of a topological vector space.
Examples
The class Ste of stereotype spaces is extremely wide, so that it will not be a serious exaggeration to say that all topological vector spaces really used in analysis are stereotype.^{[5]} Each pseudocomplete barreled space (in particular, each Banach space and each Fréchet space) is stereotype. Its dual space (which is not barreled, unless is a Montel space) is stereotype as well^{[6]}. There exist stereotype spaces which are not Mackey spaces.
Some simple connections between the properties of a stereotype space and those of its dual space are expressed in the following list of regularities:^{[7]}
- is a normed space is a Banach space is a Smith space;
- is metrizable is a Fréchet space is a Brauner space;
- is barreled has the Heine-Borel property;
- is quasi-barreled in if a set is absorbed by each barrel, then is totally bounded;
- is a Mackey space in every -weakly compact set is compact;
- is a Montel space is barreled and has the Heine-Borel property is a Montel space;
- is a space with a weak topology in every compact set is finite-dimensional;
- is separable in there is a sequence of closed subspaces of finite co-dimension with trivial intersection: .
- has the (classical) approximation property has the (classical) approximation property;
- is complete is co-complete^{[8]} is saturated;^{[9]}
- is a Pták space^{[10]} in a subspace is closed if it has the closed intersection with each compact set ;
- is hypercomplete^{[11]} in an absolutely convex set is closed if it has the closed intersection with each compact set .
Counterexamples:
- If a metrizable locally convex space is not complete, then it is not stereotype.
- Each infinite dimensional normed space considered with the -weak topology is not stereotype.
History
The first results on this type of reflexivity of topological vector spaces were obtained by M. F. Smith^{[12]}^{[13]} in 1952. Further investigations were conducted by B. S. Brudovskii, ^{[14]} W. C. Waterhouse,^{[15]} K. Brauner,^{[16]} S. S. Akbarov,^{[17]}^{[18]}^{[19]}^{[20]} and E. T. Shavgulidze.^{[21]} The term "stereotype space" was introduced by S. S. Akbarov in 1995^{[22]}. The main properties of the category of stereotype spaces were described by S. S. Akbarov in his series of works of 1995-2017.
Pseudocompletion and pseudosaturation
Each locally convex space can be transformed into a stereotype space with the help of two standard operations, pseudocompletion and pseudosaturation, defined by the following two propositions.^{[23]}
1. With any locally convex space , one can associate a linear continuous map into some pseudocomplete locally convex space , called pseudocompletion of , in such a way that the following conditions are fulfilled:
- is pseudocomplete if and only if is an isomorphism;
- for any linear continuous map of locally convex spaces, there exists a unique linear continuous map such that .
One can imagine the pseudocompletion of as the "nearest to from the outside" pseudocomplete locally convex space, so that the operation adds to some supplementary elements, but does not change the topology of (like the usual operation of completion).
2. With any locally convex space , one can associate a linear continuous map from some pseudosaturated locally convex space , called pseudosaturation of , in such a way that the following conditions are fulfilled:
- is pseudosaturated if and only if is an isomorphism;
- for any linear continuous map of locally convex spaces, there exists a unique linear continuous map such that .
The pseudosaturation of can be imagined as the "nearest to from the inside" pseudosaturated locally convex space, so that the operation strengthens the topology of , but does not change the elements of .
If is a pseudocomplete locally convex space, then its pseudosaturation is stereotype. Dually, if is a pseudosaturated locally convex space, then its pseudocompletion is stereotype. For arbitrary locally convex space the spaces and are stereotype.^{[24]}
Immediate subspaces and immediate quotient spaces
The idea of subspace (and of quotient space) in stereotype theory leads to more complicated results than in the theory of locally convex spaces.
Immediate subspaces and envelopes
The notion of immediate subspace gives a "concrete description" of the abstract notion of immediate monomorphism^{[25]}, or, what is equivalent in this situation^{[26]}, strong monomorphism in the category Ste. Surprisingly, this description does not coincide with the construction of closed subspace in the category LocConv of locally convex spaces.
- Suppose is a subset in a stereotype space endowed with a structure of a stereotype space in such a way that the set-theoretic inclusion is a morphism of stereotype spaces (i.e. a continuous linear map). Then is called a subspace of the stereotype space , with the notation
- .
- Suppose we have a chain of stereotype subspaces
- ,
- and the first mapping is a bimorphism of stereotype spaces. Then the space is called a mediator of the subspace in the space .
- A subspace in a stereotype space is called an immediate subspace in , with the notation
- ,
- if it has no non-trivial mediators, i.e. for any mediator of in the inclusion is an isomorphism.
Examples:
- An immediate subspace in a stereotype space is said to be closed, if (as a set) is closed in (as a topological space). If is a closed subspace in a stereotype space (as in a locally convex space), then its pseudosaturation is a closed immediate subspace in . All closed immediate subspaces have this form.
- There are stereotype spaces with closed immediate subspaces whose topology is not inherited from ^{[27]} (this is one of the qualitative differences with the category LocConv of locally convex spaces).^{[28]}
- In contrast to the category LocConv of locally convex spaces in the category Ste the immediate subspaces are not always closed.^{[28]}
Theorem.^{[29]} For any set in a stereotype space there is a minimal immediate subspace in , containing :
- (i)
- (ii) ,
and this subspace is an immediate subspace in each immediate subspace, containing :
- (iii) ,
- The subspace is called an envelope of the set in the stereotype space .
Theorem.^{[30]} Each set in a stereotype space is a total set^{[31]} in its envelope .
If denotes the space of all functions with finite support, endowed with the strongest locally convex topology, and the mapping acts by the formula , then the envelope coincides with the abstract categorical envelope of the space in the class of all epimorphisms in the category Ste with respect to the morphism :^{[32]}
Immediate quotient spaces and refinements
Dually, the notion of immediate quotient space gives a "concrete description" of the abstract notion of immediate epimorphism^{[33]}, or, what is equivalent here^{[26]}, strong epimorphism in the category Ste. Like in the situation with monomorphisms, this description does not coincide with the construction of quotient space in the category LocConv of locally convex spaces.
- Let be a closed subspace (in the usual sense) in a stereotype space . Consider a topology on the quotient space , which is majorized by the usual quotient topology of . Let be a completion of with respect to the topology . Suppose is a subset in the locally convex space which contains and at the same time is a stereotype space. Then is called a quotient space of the stereotype space , with the notation
- .
- Suppose we have two quotient spaces and . It is said that the subordinates (notation: ) if there is a morphism such that (where and are the natural mappings).
- Suppose that the quotient space subordinates the quotient space (i.e. ) and the corresponding morphism is a bimorphism. Then the quotient space is called a mediator of the quotient space of the space .
- A quotient space of a stereotype space is called an immediate quotient space of , with the notation
- ,
- if it has no non-trivial mediators, i.e. for any mediator of the morphism is an isomorphism.
Examples:
- An immediate quotient space of a stereotype space is said to be open, if the corresponding map is open^{[34]}. If is a closed subspace in a stereotype space , then the pseudocompletion of the (locally convex) quotient space is an open immediate quotient space of . All open immediate quotient spaces have this form.
- There are stereotype spaces with immediate quotient spaces which cannot be represented in the form .^{[35]}
- In contrast to the category LocConv of locally convex spaces in the category Ste immediate quotient spaces are not always open.^{[35]}
Theorem.^{[36]} For any set of linear continuous functionals on a stereotype space there is a minimal immediate quotient space of to which all functionals can be extended:
- (i)
- (ii) ,
and this quotient space is (up to an isomorphism) an immediate quotient space of each immediate quotient space, to which the functionals are extended:
- (iii) .
- The quotient space is called a refinement of the set on the stereotype space .
Theorem.^{[37]} Each set of linear continuous functionals on a stereotype space is a total set^{[38]} on its refinement .
If denotes the space of all functions , endowed with the topology of pointwise convergence, and the mapping acts by the formula , then the refinement coincides with the abstract categorical refinement of the space in the class of all monomorphisms in the category Ste by means of the morphism :^{[32]}
Category Ste of stereotype spaces
The class Ste of stereotype spaces forms a category with linear continuous maps as morphisms and possesses the following properties:
- Ste is pre-abelian^{[39]};
- Ste is bicomplete^{[40]};
- Ste is autodual with respect to the functor ^{[41]};
- Ste is a category with nodal decomposition^{[42]}.
- Ste is a *-autonomous category.^{[43]}
Kernel and cokernel in the category Ste
Ste is a pre-abelian category: each morphism in the category Ste has a kernel
and a cokernel
As a corollary, has an image and a coimage as well. The following natural identities hold:^{[39]}
where denotes the pseudosaturation of the annihilator of the subspace in the dual space :
- .
Ste as a *-autonomous category
For any two stereotype spaces and the stereotype space of operators from into , is defined as the pseudosaturation of the space of all linear continuous maps endowed with the topology of uniform convergeance on totally bounded sets. The space is stereotype. It defines two natural tensor products
Theorem. In the category Ste the following natural identities hold:^{[43]}^{[44]}:
In particular, Ste is a symmetric monoidal category with respect to the bifunctor , a symmetric closed monoidal category with respect to the bifunctor and the internal hom-functor , and a *-autonomous category:
Limits and colimits in the category Ste
Ste is a bicomplete category: each small diagram Ste has a limit, , which coincides with the pseudocompletion of the corresponding limit in the category LocConv of locally convex spaces
- ,
and a colimit, , which coincides with the pseudosaturation of the corresponding colimit in LocConv
- .
However, the direct sum and the direct product in Ste coincide with the corresponding constructions in LocConv:
The following natural identities hold:^{[43]}^{[44]}
Grothendieck transformation
If and are stereotype spaces then for each elements and the formula
defines an elementary tensor , and the formula
defines an elementary tensor
Theorem.^{[45]} For each stereotype spaces and there is a unique linear continuous map which turns elementary tensors into elementary tensors :
The family of maps defines a natural transformation of the bifunctor into the bifunctor .
- The map is called the Grothendieck transformation.
Stereotype approximation property
A stereotype space is said to have the stereotype approximation property, if each linear continuous map can be approximated in the stereotype space of operators by the linear continuous maps of finite rank. This condition is weaker than the existence of the Schauder basis, but formally stronger than the classical approximation property (however, it is not clear (2017) whether the stereotype approximation property coincides with the classical one, or not).
Theorem.^{[46]} For a stereotype space the following conditions are equivalent:
- (i) has the stereotype approximation property;
- (ii) the Grothendieck transformation is a monomorphism (in the category Ste);
- (iii) the Grothendieck transformation is an epimorphism (in the category Ste);
- (iv) for any stereotype space the Grothendieck transformation is a monomorphism (in the category Ste);
- (v) for any stereotype space the Grothendieck transformation is an epimorphism (in the category Ste).
Theorem.^{[47]} If two stereotype spaces and have the stereotype approximation property, then the spaces , and have the stereotype approximation property as well.
In particular, if has the stereotype approximation property, then the same is true for and for .
Universality of
For any stereotype spaces , , a bilinear map is said to be continuous (as a bilinear map of stereotype spaces) if
- 1) for each neighborhood of zero and for each compact set there exists a neighborhood of zero such that , and
- 2) for each neighborhood of zero and for each compact set there exists a neighborhood of zero such that .
Examples:
- For any stereotype space the pairing is a continuous bilinear map.
- For any two stereotype spaces and the map is a continuous bilinear map.
- For any two stereotype spaces and the map is a continuous bilinear map.
Theorem.^{[48]} For any stereotype spaces , , and for any continuous bilinear map there exists a unique continuous linear map such that , where .
Corollary.^{[47]} For any stereotype space the pairing has a unique extension to a linear continuous functional . This functional in its turn can be represented as a trace of the operators occurring as images of the tensors under the Grothendieck transformation if and only if the space has the stereotype approximation property.
Applications
Being a symmetric monoidal category, Ste generates the notions of a stereotype algebra (as a monoid in Ste) and a stereotype module (as a module in Ste over such a monoid), and for each stereotype algebra the categories _{$A$}Ste and Ste_{$A$} of left and right stereotype modules over are enriched categories over Ste.^{[49]} This distinguishes the category Ste from the other known categories of locally convex spaces, since up to the recent time only the category Ban of Banach spaces and the category Fin of finite-dimensional spaces had been known to possess this property. On the other hand, the category Ste is so wide, and the tools for creating new spaces in Ste are so diverse, that this suggests the idea that all the results of functional analysis can be reformulated inside the stereotype theory without essential losses. On this way one can even try to completely replace the category of locally convex spaces in analysis (and in related areas) by the category Ste of stereotype spaces with the view of possible simplifications – this program was announced by S. Akbarov in 2005^{[5]} and the following results can be considered as evidences of its reasonableness:
- In the theory of stereotype spaces the approximation property is inherited by the spaces of operators and by tensor products. This allows to reduce the list of counterexamples in comparison with the Banach theory, where as is known the space of operators does not inherit the approximation property.^{[50]}
- The arising theory of stereotype algebras allows to simplify constructions in the duality theories for non-commutative groups. In particular, the group algebras (and their envelopes in the necessary cases) in these theories become Hopf algebras in the standard algebraic sense.^{[51]}^{[52]}^{[53]}^{[20]}
- This in its turn leads to a family of generalizations of the Pontryagin duality based on the notion of envelope: the holomorphic, the smooth and the continuous envelopes of stereotype algebras give rise respectively to the holomorphic, the smooth and the continuous dualities in big geometric disciplines – complex geometry, differential geometry, and topology – for certain classes of (not necessarily commutative) topological groups considered in these disciplines (affine algebraic groups, and some classes of Lie groups and Moore groups).^{[18]}^{[19]}^{[20]}^{[53]}
See also
- Stereotype algebra
- Stereotype group algebra
- Reflexive space
- Duality (mathematics)
- Pontryagin duality
- Envelope (category theory)
- Refinement (category theory)
- Brauner space
- Smith space
Notes
- ^ Akbarov 2003, p. 219.
- ^ ...or over the field of real numbers, with the similar definition.
- ^ Akbarov 2003, p. 219, 220.
- ^ A set is said to be capacious if for each totally bounded set there is a finite set such that .
- ^ ^{a} ^{b} Akbarov 2005.
- ^ Of course, this is a general fact: if is stereotype then is also stereotype.
- ^ Akbarov 2003, p. 221.
- ^ A locally convex space is called co-complete if each linear functional which is continuous on every totally bounded set , is automatically continuous on the whole space .
- ^ A locally convex space is said to be saturated if for an absolutely convex set being a neighbourhood of zero in is equivalent to the following: for each totally bounded set there is a closed neighbourhood of zero in such that .
- ^ A locally convex space is called a Pták space, or a fully complete space, if in its dual space a subspace is -weakly closed when it has -weakly closed intersection with the polar of each neighbourhood of zero .
- ^ A locally convex space is said to be hypercomplete if in its dual space every absolutely convex space is -weakly closed if it has -weakly closed intersection with the polar of each neighbourhood of zero .
- ^ Smith 1952.
- ^ Onishchik 1984.
- ^ Brudovski 1967.
- ^ Waterhouse 1968.
- ^ Brauner 1973.
- ^ Akbarov 2003.
- ^ ^{a} ^{b} Akbarov 2009.
- ^ ^{a} ^{b} Akbarov 2016.
- ^ ^{a} ^{b} ^{c} Akbarov 2017.
- ^ Akbarov & Shavgulidze 2003.
- ^ Akbarov 1995.
- ^ Akbarov 2003, p. 197, 200.
- ^ It is not clear (2017) whether and coincide.
- ^ A monomorphism is said to be immediate if in each representation , where is a monomorphism and is an epimorphism, the morphism is automatically an isomorphism.
- ^ ^{a} ^{b} Akbarov 2016, p. 39.
- ^ In other words, in this case the topology of inherited from is not pseudosaturated.
- ^ ^{a} ^{b} Akbarov 2016, p. 128.
- ^ Akbarov 2016, p. 134.
- ^ Akbarov 2016, p. 131.
- ^ I.e. the linear span of is dense in (as in a locally convex space).
- ^ ^{a} ^{b} Akbarov 2016, p. 144.
- ^ An epimorphism is said to be immediate if in each representation , where is a monomorphism and is an epimorphism, the morphism is automatically an isomorphism.
- ^ A linear map is said to be open, if for each neighborhood of zero there is a neighborhood of zero such that .
- ^ ^{a} ^{b} Akbarov 2016, p. 138.
- ^ Akbarov 2016, p. 140.
- ^ This proposition is dual to the corresponding theorem on envelopes in stereotype spaces.
- ^ I.e. if , then there exists such that .
- ^ ^{a} ^{b} Akbarov 2003, p. 224.
- ^ Akbarov 2003, p. 226.
- ^ Akbarov 2003, p. 220.
- ^ Akbarov 2016, p. 142.
- ^ ^{a} ^{b} ^{c} Akbarov 2003, p. 245.
- ^ ^{a} ^{b} Akbarov 2017, p. 581.
- ^ Akbarov 2003, p. 246.
- ^ Akbarov 2003, p. 264.
- ^ ^{a} ^{b} Akbarov 2003, p. 265.
- ^ Akbarov 2003, p. 242.
- ^ Akbarov 2003, p. 289.
- ^ Szankowski 1981.
- ^ Akbarov 2003, p. 278.
- ^ Akbarov 2009, p. 507.
- ^ ^{a} ^{b} Kuznetsova 2013.
External links
- Stereotype space in nLab
References
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