Editing Reflexive space (section)

== Other types of reflexivity ==

A stereotype space, or polar reflexive space, is defined as a [[topological vector space]] (TVS) satisfying a similar condition of reflexivity, but with the topology of uniform convergence on [[Totally bounded set|totally bounded]] subsets (instead of [[Bounded set|bounded]] subsets) in the definition of dual space <math>X^{\prime}.</math> More precisely, a TVS <math>X</math> is called polar reflexive<ref>{{cite book|last=Köthe|first=Gottfried|title=Topological Vector Spaces I|publisher=Springer|series=Springer Grundlehren der mathematischen Wissenschaften|year=1983|url=https://www.springer.com/gp/book/9783642649905#aboutBook|isbn=978-3-642-64988-2 }}</ref> or stereotype if the evaluation map into the second dual space
<math display="block">J : X \to X^{\star\star},\quad J(x)(f) = f(x),\quad x\in X,\quad f\in X^\star</math>
is an [[TVS-isomorphism|isomorphism of topological vector spaces]].<ref name=isomorphism /> Here the stereotype dual space <math>X^\star</math> is defined as the space of continuous linear functionals <math>X^{\prime}</math> endowed with the topology of uniform convergence on totally bounded sets in <math>X</math> (and the ''stereotype second dual space'' <math>X^{\star\star}</math> is the space dual to <math>X^{\star}</math> in the same sense).

In contrast to the classical reflexive spaces the class '''Ste''' of stereotype spaces is very wide (it contains, in particular, all [[Fréchet space]]s and thus, all [[Banach space]]s), it forms a [[closed monoidal category]], and it admits standard operations (defined inside of '''Ste''') of constructing new spaces, like taking closed subspaces, quotient spaces, projective and injective limits, the space of operators, tensor products, etc. The category '''Ste''' have applications in duality theory for non-commutative groups.

Similarly, one can replace the class of bounded (and totally bounded) subsets in <math>X</math> in the definition of dual space <math>X^{\prime},</math> by other classes of subsets, for example, by the class of compact subsets in <math>X</math> – the spaces defined by the corresponding reflexivity condition are called {{em|reflective}},<ref>{{cite journal|last=Garibay Bonales|first=F.|author2=Trigos-Arrieta, F. J.|author3=Vera Mendoza, R.|title=A characterization of Pontryagin-van Kampen duality for locally convex spaces|journal=Topology and Its Applications|year=2002|volume=121|issue=1–2 |pages=75–89|doi=10.1016/s0166-8641(01)00111-0|doi-access=free}}</ref><ref>{{cite journal|last=Akbarov|first=S. S.|author2=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}}</ref> and they form an even wider class than '''Ste''', but it is not clear (2012), whether this class forms a category with properties similar to those of '''Ste'''.