Every Banach Space is Reflexive

Abstract

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. This, from a categorical point of view, is indeed the right duality concept because it yields a self adjoint dualisation functor. However, for many applications the non-reflexiveness problem can be solved by replacing the norm on the first dual by the weak*-structure [1].But then, by taking the second dual, only the original vector space is recovered and no universal property remains with this modified dual structure. In this work we unify the applied and the structural point of view.We introduce a suitable numerical structure on vector spaces such that Banach balls, or more precisely totally convex modules, arise naturally in duality, i.e. 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.