Stereotype Spaces and Other Versions of Reflexivity
Among all locally convex spaces (even among all Banach spaces) used in functional analysis the class of reflexive spaces is too narrow to represent a self-sufficient category in any sense. On the other hand, the idea of duality reflected in this notion is so natural that it gives rise to intuitive expectations that appropriate changes in the definition of reflexivity can lead to another notion, more convenient for some goals of mathematics. One of such goals is the idea of approaching analysis to the other parts of mathematics, like algebra and geometry, by reformulating its results in the purely algebraic language of category theory.
This program is being worked out in the theory of stereotype spaces, which are defined as topological vector spaces satisfying a similar condition of reflexivity, but with the topology of uniform convergence on totally bounded subsets (instead of bounded subsets) in the definition of dual space X’. In contrast to the classical reflexive spaces the class Ste of stereotype spaces is very wide (it contains, in particular, all Fréchet spaces and thus, all Banach spaces), 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 X in the definition of dual space X’, by other classes of subspaces, for example, by the class of compact subsets in X -- the spaces defined by the corresponding reflexivity condition are called reflective, 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.
Read more about this topic: Reflexive Space
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