en>Michael Hardy: closer to the norms of WP:MOS and WP:MOSMATH

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In [[functional analysis]] and related areas of [[mathematics]] '''stereotype spaces''' are [[topological vector space]]s defined by a special variant of [[Reflexive space|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 space]]s and thus, all [[Banach space]]s), it consists of spaces satisfying a natural condition of completeness, and it forms a [[closed monoidal category]] with the standard analytical tools for constructing new spaces, like taking closed subspace, quotient space, projective and injective limits, the space of operators, tensor products, etc.
[[File:Stereotype spaces.jpg|thumbnail|Mutual embeddings of the main classes of locally convex spaces]]

==Definition==

A '''stereotype space'''<ref name=Akbarov-1>{{harvtxt|S.S.Akbarov|2003}}.</ref> is a [[topological vector space]] <math>X</math> over the field <math>\mathbb{C}</math> of complex numbers<ref>...or over the field <math>\mathbb{R}</math> of real numbers, with the similar definition.</ref> such that the natural map into the second dual space

:<math> i:X\to X^{\star\star},\quad i(x)(f)=f(x),\quad x\in X,\quad f\in X^\star </math>

is an isomorphism of topological vector spaces (i.e. a [[Linear map|linear]] and a [[Homeomorphism|homeomorphic]] map). Here the ''dual space'' <math> X^\star</math> is defined as the space of all linear continuous functionals <math>f:X\to\mathbb{C}</math> endowed with the topology of uniform convergence on [[totally bounded set]]s in ''X'', and the ''second dual space'' <math> X^{\star\star}</math> is the space dual to <math> X^{\star}</math> in the same sense.

The following criterion holds:<ref name=Akbarov-1 /> a topological vector space <math>X</math> is stereotype if and only if it is [[Locally convex space|locally convex]] and satisfies the following two conditions:

:* ''pseudocompleteness'': each [[Totally bounded set|totally bounded]] Cauchy net in <math>X</math> converges,
:*
:* ''pesudosaturateness'': each closed convex balanced ''capacious''<ref>A set <math>D\subseteq X</math> is said to be ''capacious'' if for each [[Totally bounded set|totally bounded]] set <math>A\subseteq X</math> there is a finite set <math>F\subseteq X</math> such that <math>A\subseteq D+F</math>.</ref> set <math>D</math> in <math>X</math> is a neighborhood of zero in <math>X</math>.

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 [[Barreled space|barreledness]] of a topological vector space.

==Examples==

Each pseudocomplete [[barreled space]] <math>X</math> (in particular, each [[Banach space]] and each [[Frechet space|Fréchet space]]) is stereotype. A metrizable locally convex space <math>X</math> is stereotype if and only if <math>X</math> is complete. A normed space <math>X</math> with the <math>X^{\star}</math>-weak topology is stereotype if and only if X has finite dimension. There exist stereotype spaces which are not [[Mackey space]]s.

Some simple connections between the properties of a stereotype space <math>X</math> and those of its dual space <math>X^\star</math> are expressed in the following list of regularities.<ref name=Akbarov-1 /><ref name=Akbarov-2>{{harvtxt|S.S.Akbarov|2009}}.</ref> For a stereotype space <math>X</math>

:* <math>X</math> is a [[normed space]] <math>\Longleftrightarrow</math> <math>X</math> is a [[Banach space]] <math>\Longleftrightarrow</math> <math>X^\star</math> is a [[Smith space]];

:* <math>X</math> is metrizable <math>\Longleftrightarrow</math> <math>X</math> is a [[Fréchet space]] <math>\Longleftrightarrow</math> <math>X^\star</math> is a [[Brauner space]];

:* <math>X</math> is [[barreled space|barreled]] <math>\Longleftrightarrow</math> <math>X^\star</math> has the Heine-Borel property;

:* <math>X</math> is quasibarreled <math>\Longleftrightarrow</math> in <math>X^\star</math> if a set <math>T</math> is absorbed by each barrel, then <math>T</math> is totally bounded;

:* <math>X</math> is a [[Mackey space]] <math>\Longleftrightarrow</math> in <math>X^\star</math> every <math>X</math>-weakly compact set is compact;

:* <math>X</math> is a [[Montel space]] <math>\Longleftrightarrow</math> <math>X</math> is barreled and has the Heine-Borel peoperty <math>\Longleftrightarrow</math> <math>X^\star</math> is a Montel space;

:* <math>X</math> is a space with a weak topology <math>\Longleftrightarrow</math> in <math>X^\star</math> every compact set <math>T</math> is finite-dimentional;

:* <math>X</math> is [[separable space|separable]] <math>\Longleftrightarrow</math> in <math>X^\star</math> there is a sequence of closed subspaces <math>L_n</math> of finite co-dimension with trivial intersection: <math>\bigcap_{n=1}^{\infty}L_n=\{0\}</math>.

:* <math>X</math> has the (classical) [[approximation property]] <math>\Longleftrightarrow</math> <math>X^\star</math> has the (classical) [[approximation property]];

:* <math>X</math> is [[Uniform space|complete]] <math>\Longleftrightarrow</math> <math>X^\star</math> is co-complete<ref>A locally convex space <math>X</math> is called ''co-complete'' if each linear functional <math>f:X\to\mathbb{C}</math> which is continuous on every totally bounded set <math>S\subseteq X</math>, is automatically continuous on the whole space <math>X</math>.</ref> <math>\Longleftrightarrow</math> <math>X^\star</math> is saturated;<ref>A locally convex space <math>X</math> is said to be ''saturated'' if for an absolutely convex set <math>B\subseteq X</math> being a neighbourhood of zero in <math>X</math> is equivalent to the following: for each totally bounded set <math>S\subseteq X</math> there is a closed neighbourhood of zero <math>U</math> in <math>X</math> such that <math>B\cap S=U</math>.</ref>

:* <math>X</math> is a Pták space<ref>A locally convex space <math>X</math> is called a ''Pták space'', or a '''fully complete space''', if in its dual space <math>X^\star</math> a subspace <math>Q\subseteq X^\star</math> is <math>X</math>-weakly closed when it has <math>X</math>-weakly closed intersection with the polar <math>U^\circ</math> of each neighbourhood of zero <math>U\subseteq X</math>.</ref> <math>\Longleftrightarrow</math> in <math>X^\star</math> a subspace <math>L</math> is closed if it has clodes intersection <math>L\cap K</math> with each compact set <math>K\subseteq X^\star</math>;

:* <math>X</math> is hypercomplete<ref>A locally convex space <math>X</math> is said to be ''hypercomplete'' if in its dual space <math>X^\star</math> every absolutely convex space <math>Q\subseteq X^\star</math> is <math>X</math>-weakly closed if it has <math>X</math>-weakly closed intersection with the polar <math>U^\circ</math> of each neighbourhood of zero <math>U\subseteq X</math>.</ref> <math>\Longleftrightarrow</math> in <math>X^\star</math> an absolutely convex set <math>B</math> is closed if it has the closed intersection <math>B\cap K</math> with each compact set <math>K\subseteq X^\star</math>.

==History==

The first results on this type of reflexivity of topological vector spaces were obtained by M. F. Smith<ref name=Smith>{{harvtxt|M.F.Smith|1952}}.</ref> in 1952. Further investigations were conducted by B. S. Brudovskii,
<ref name=Brudovski>{{harvtxt|B.S.Brudowski|1967}}.</ref> W. C. Waterhouse,<ref name=Waterhouse>{{harvtxt|W.C.Waterhouse|1968}}.</ref> K. Brauner,<ref name=Brauner>{{harvtxt|K.Brauner|1973}}.</ref> S. S. Akbarov,<ref name=Akbarov-1 /><ref name=Akbarov-2 /><ref name=Akbarov-3>{{harvtxt|S.S.Akbarov|2013}}.</ref> and E. T. Shavgulidze.<ref name=Akbarov-Shavgulidze>{{harvtxt|S.S.Akbarov, E.T.Shavgulidze|2003}}.</ref>

==Pseudocompletion and pseudosaturation==

Each [[locally convex space]] <math>X</math> can be transformed into a stereotype space with the help of the standard operations of pseudocompletion and pseudosaturation defined by the following two propositions.<ref name=Akbarov-1 />

1. With any locally convex space <math>X</math>, one can associate a linear continuous map <math>\triangledown_X: X\to X^\triangledown</math> into some pseudocomplete locally convex space <math>X^\triangledown</math>, called ''pseudocompletion'' of <math>X</math>, in such a way that the following conditions are fulfilled:
:* <math>X</math> is pseudocomplete if and only if <math>\triangledown_X: X\to X^\triangledown</math> is an isomorphism;
:* for any linear continuous map <math>\varphi:X\to Y</math> of locally convex spaces, there exists a unique linear continuous map <math>\varphi^\triangledown:X^\triangledown\to Y^\triangledown</math> such that <math>\triangledown_Y\circ\varphi=\varphi^\triangledown\circ\triangledown_X</math>.

One can imagine the pseudocompletion of <math>X</math> as the "nearest to <math>X</math> from the outside" pseudocomplete locally convex space, so that the operation <math>X\mapsto X^\triangledown</math> adds to <math>X</math> some supplementary elements, but does not change the topology of <math>X</math> (like the usual operation of completion).

2. With any locally convex space <math>X</math>, one can associate a linear continuous map <math>\vartriangle_X:X^\vartriangle\to X</math> from some pseudosaturated locally convex space <math>X^\vartriangle</math>, called ''pseudosaturation'' of <math>X</math>, in such a way that the following conditions are fulfilled:
:* <math>X</math> is pseudosaturated if and only if <math>\vartriangle_X:X^\vartriangle\to X</math> is an isomorphism;
:* for any linear continuous map <math>\varphi:X\to Y</math> of locally convex spaces, there exists a unique linear continuous map <math>\varphi^\vartriangle:X^\vartriangle\to Y^\vartriangle</math> such that <math>\varphi\circ\vartriangle_X=\vartriangle_Y\circ\varphi^\vartriangle</math>.

The pseudosaturation of <math>X</math> can be imagined as the "nearest to <math>X</math> from the inside" pseudosaturated locally convex space, so that the operation <math>X\mapsto X^\vartriangle</math> strengthen the topology of <math>X</math>, but does not change the elements of <math>X</math>.

If <math>X</math> is a pseudocomplete locally convex space, then its pseudosaturation <math>X^\vartriangle</math> is stereotype. Dually, if <math>X</math> is a pseudosaturated locally convex space, then its pseudocompletion <math>X^\triangledown</math> is stereotype. For arbitrary locally convex space <math>X</math> the spaces <math>X^{\vartriangle\triangledown}</math> and <math>X^{\triangledown\vartriangle}</math> are stereotype.<ref>It is not clear (2013) whether <math>X^{\vartriangle\triangledown}</math> and <math>X^{\triangledown\vartriangle}</math> coincide.</ref>

==Category of stereotype spaces==

The class '''Ste''' of stereotype spaces forms a category with linear continuous maps as morphisms and has the following properties:,<ref name=Akbarov-1 /><ref name=Akbarov-3 />

:* '''Ste''' is [[Pre-abelian category|pre-abelian]];
:*
:* '''Ste''' is [[Complete category|complete]] and [[Complete category|co-complete]];
:*
:* '''Ste''' is autodual with respect to the functor <math>X\to X^\star</math> of passing to the dual space;
:*
:* '''Ste''' is a category with ''nodal decomposition'': each morphism <math>\varphi:X\to Y</math> has a decomposition <math>\varphi=\sigma\circ\beta\circ\pi</math>, where <math>\pi</math> is a strong epimorphism, <math>\beta</math> a bimorphism, and <math>\sigma</math> a strong monomorphism.

For any two stereotype spaces <math>X</math> and <math>Y</math> the ''stereotype space of operators'' <math>\text{Hom}(X,Y)</math> from <math>X</math> into <math>Y</math>, is defined as the pseudosaturation of the space <math>\text{L}(X,Y)</math> of all linear continuous maps <math>\varphi:X\to Y</math> endowed with the topology of uniform convergeance on totally bounded sets. The space <math>\text{Hom}(X,Y)</math> is stereotype. It defines two natural tensor products

: <math>X\circledast Y:= \text{Hom}(X,Y^\star)^\star,</math>

:<math> X\odot Y := \text{Hom}(X^\star,Y).</math>

The following natural identities hold:<ref name=Akbarov-1 />

:<math>\mathbb{C}\circledast X\cong X\cong X\circledast \mathbb{C},
</math>

:<math>
\mathbb{C}\odot X\cong X\cong X\odot\mathbb{C},
</math>

:<math> X\circledast Y\cong Y\circledast X,</math>

:<math>
X\odot Y\cong Y\odot X,
</math>

:<math> (X\circledast Y)\circledast Z\cong X\circledast (Y\circledast Z),</math>

:<math>
(X\odot Y)\odot Z\cong X\odot (Y\odot Z),
</math>

:<math> (X\circledast Y)^\star\cong Y^\star\odot X^\star,</math>

:<math>
(X\odot Y)^\star\cong Y^\star\circledast X^\star,
</math>

:<math>
\text{Hom}(X\circledast Y,Z)\cong \text{Hom}(X,\text{Hom}(Y,Z)),</math>

:<math>
\text{Hom}(X,Y\odot Z)\cong \text{Hom}(X,Y)\odot Z
</math>

As a corollary,
:* '''Ste''' is a symmetric [[monoidal category]] with respect to the bifunctor <math>\odot</math> and a symmetric [[closed monoidal category]] with respect to the bifunctor <math>\circledast</math> and the internal hom-functor <math>\text{Hom}</math>.

==Stereotype approximation property==

A stereotype space <math>X</math> is said to have the ''stereotype approximation property'', if each linear continuous map <math>\varphi:X\to X</math> can be approximated in the stereotype space of operators <math>\text{Hom}(X,X)</math> 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 (2013) whether the stereotype approximation property coincide with the classical one, or not). The following proposition holds:

:* If two stereotype spaces <math>X</math> and <math>Y</math> have the stereotype approximation property, then the spaces <math>\text{Hom}(X,Y)</math>, <math>X\circledast Y</math> and <math>X\odot Y</math> have the stereotype approximation property as well.<ref name=Akbarov-1 />

In particular, if <math>X</math> has the stereotype approximation property, then the same is true for <math>X^\star</math> and for <math>\text{Hom}(X,X)</math>.

==Applications==

Being a symmetric monoidal category, '''Ste''' generates the notions of a ''stereotype algebra'' (as a [[Monoid (category theory)|monoid]] in '''Ste''') and a ''stereotype module'' (as a module in '''Ste''' over such a monoid), and for each stereotype algebra <math>A</math> the categories <sub><math>A</math></sub>'''Ste''' and '''Ste'''<sub><math>A</math></sub> of left and right stereotype modules over <math>A</math> are [[Enriched category|enriched categories]] over '''Ste'''.<ref name=Akbarov-1 /> 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 functional 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<ref name=Akbarov-4>{{harvtxt|S.S.Akbarov|2005}}.</ref> 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.<ref name=Szankowski>{{harvtxt|A.Szankowski|1981}}.</ref>
:*
:* The arising theory of stereotype algebras allows to simplify constructions in the duality theories for non-commutative groups. In particular, the group algebras in these theories become [[Hopf algebra]]s in the standard algebraic sense.<ref name=Akbarov-2 /><ref name=Kuznetsova>{{harvtxt|J.Kuznetsova|2013}}.</ref>

==Notes==
{{reflist}}

==References==
* {{cite book
| last = Schaefer
| first = Helmuth H. 
| year = 1966
| title = Topological vector spaces
| series=
| volume=
| publisher = The MacMillan Company
| location = New York
| isbn = 0-387-98726-6
}}

* {{cite book |last=Robertson |first=A.P. |coauthors=Robertson, W.J. |title= Topological vector spaces |series=Cambridge Tracts in Mathematics |volume=53 |year=1964 |publisher= [[Cambridge University Press]] }}

*{{cite journal|last=Smith|first=M.F.|title=[http://www.jstor.org/stable/1969798 The Pontrjagin duality theorem in linear spaces]|journal=Annals of Mathematics|year=1952|volume=56|issue=2|pages=248–253|doi=10.2307/1969798}}

*{{cite journal|last=Brudovski|first=B.S.|title=On k- and c-reflexivity of locally convex vector spaces|journal=Lithuanian Mathematical Journal|year=1967|volume=7|issue=1|pages=17–21}}

*{{cite journal|last=Waterhouse|first=W.C.|title=[http://projecteuclid.org/DPubS?verb=Display&version=1.0&service=UI&handle=euclid.pjm/1102986038&page=record Dual groups of vector spaces]|journal=Pac. J. Math.|year=1968|volume=26|issue=1|pages=193–196|doi=10.2140/pjm.1968.26.193}}

*{{cite journal|last=Brauner|first=K.|title=Duals of Frechet spaces and a generalization of the Banach-Dieudonne theorem|journal=Duke Math. Jour.|year=1973|volume=40|issue=4|pages=845–855|doi=10.1215/S0012-7094-73-04078-7}}

*{{cite journal|last=Akbarov|first=S.S.|title=[http://www.springerlink.com/content/k62m72960101g6q2/ Pontryagin duality in the theory of topological vector spaces and in topological algebra]|journal=Journal of Mathematical Sciences|year=2003|volume=113|issue=2|pages=179–349|doi=10.1023/A:1020929201133}}

*{{cite journal|last=Akbarov|first=S.S.|title=[http://www.springerlink.com/content/u07317731010573l/ Holomorphic functions of exponential type and duality for Stein groups with algebraic connected component of identity]{{subscription required}}|journal=Journal of Mathematical Sciences|year=2009|volume=162|issue=4|pages=459–586|doi=10.1007/s10958-009-9646-1}}

*{{cite arXiv|last=Akbarov|first=S.S.|title=Envelopes and imprints in categories|year=1970|eprint=1110.2013v7|class=math.FA}}

*{{cite journal|last=Akbarov|first=S.S.|coauthors=Shavgulidze, E.T.|title=On two classes of spaces reflexive in the sense of Pontryagin|journal=Mat. Sbornik|year=2003|volume=194|issue=10|pages=3–26}}

*{{cite journal|last=Kuznetsova|first=J.|title=A duality for Moore groups|journal=Journal of Operator Theory|year=2013|volume=69|issue=2|pages=101–130|arxiv=0907.1409|bibcode=2009arXiv0907.1409K|doi=10.7900/jot.2011mar17.1920}}

*{{cite journal|last=Akbarov|first=S.S.|title=[http://webmail.impan.gov.pl/cgi-bin/bc/pdf?bc67-0-05 Pontryagin duality and topological algebras]|journal=Banach Center Publications|year=2005|volume=67|pages=55–71}}

*{{cite journal|last=Szankowski|first=A.|title=B(H) does not have the approximation property|journal=Act. Math.|year=1981|volume=147|pages=147:89–108}}





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en>Michael Hardy: closer to the norms of WP:MOS and WP:MOSMATH