Resumen de Every banach space is reflexive
Christophe Van Olmen, Stijn Verwulgen
The title above is wrong, because the strong dual of a Banach space is too strong to assert that the natural correspondence between a space and its bidual is an isomorphism. However, for many applications it suffices to replace the norm on the first dual by the weak*-structure in order to solve the non-reflexiveness problem [1]. But in this way, only the original vector space is recovered by taking the second dual. In this work we introduce a suitable numerical structure on vector spaces such that Banach balls, or more precisely totally convex modules, arise naturally in duality, namely as a category of Eilenberg¿Moore algebras. This numerical structure naturally overlies the weak*-topology on the algebraic dual, so the entire Banach space can be reconstructed as a second dual. Moreover, the isomorphism between the original space and its bidual is the unit of an adjunction between the two-dualisation functors. Notice that the weak*-topology is normable only if it lives on a finite dimensional space; in that case the original space is trivial as well, hence reflexive. So the overlying numerical structure should be something more general than a norm or a seminorm and thus approach theory [2, 3] enters the picture.
