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The setting of approach frames is an exquisite environment to compare the various notions of regularity in topology, frames and approach spaces and in this paper we make this investigation. This brings to light some interesting similarities and dierences and proves that there are at least two dierent unifying notions in the setting of approach frames each with specic advantages. * The third author is an Aspirant of the Foundation for Scientic Research Flanders.

In [2] the subconstruct Sob of sober approach spaces was introduced and it was shown to be a reflective subconstruct of the category Ap 0 of T 0 approach spaces. The main result of this paper states that moreover Sob is firmly U-reflective in Ap 0 for the class U of epimorphic embeddings. 'Firm U-reflective' is a notion introduced in [3] by G.C.L. Brümmer and E. Giuli and is inspired by the exemplary behaviour of the usual completion in the category Unif 0 of Hausdorff uniform spaces with uniformly continuous maps. It means that Sob is U-reflective in Ap 0 and that the reflector is such that f : X → Y belongs to U if and only if ( f ) is an isomorphism. Firm U-reflectiveness implies uniqueness of completion in the sense that whenever f : X → Y is a map with f ∈ U and Y sober, the associated f * : (X ) → Y is an isomorphism. Our result generalizes the fact that in the category Top 0 the subconstruct of sober topological spaces is firmly reflective for the class U b of b-dense embeddings in Top 0 . Also firmness in some other subconstructs of Ap 0 will be easily obtained.

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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.