The Hahn-Banach theorem in type theory

Abstract

We present the basic concepts and definitions needed in a pointfree approach to functional analysis via formal topology. Our main results are the constructive proofs of localic formulations of the Alaoglu and Helly-Hahn-Banach theorems. Earlier pointfree formulations of the Hahn-Banach theorem, in a topos-theoretic setting, were presented by Mulvey and Pelletier (1987, 1991) and by Vermeulen (1986). A constructive proof based on points was given by Bishop (1967). In the formulation of his proof, the norm of the linear functional is preserved to an arbitrary degree by the extension and a counterexample shows that the norm, in general, is not preserved exactly. As usual in pointfree topology, our guideline is to define the objects under analysis as formal points of a suitable formal space. After this has been accomplished for the reals, we consider the formal topology ℒ(A) obtained as follows. To the formal space of mappings from a normed vector space A to the reals, we add the linearity and norm conditions in the form of covering axioms. The linear functional of norm ≤1 from A to the reals then correspond to the formal points of this formal topology. Given a subspace M of A, the classical Helly-Hahn-Banach theorem states that the restriction mapping from the linear functionals on A of norm ≤1 to those on M is surjective. In terms of covers, conceived as deductive systems, it becomes a conservativity statement (cf. Mulvey and Pelletier 1991): whenever a is an element and U is a subset of the base of the formal space ℒ(M) and we have a derivation in ℒ(A) of a ⊲ U, then we can find a derivation in ℒ(M) with the same conclusion. With this formulation it is quite natural to look for a proof by induction on covers. Moreover, as already pointed out by Mulvey and Pelletier (1991), it is possible to simplify the problem greatly, since it is enough to prove it for coherent spaces of which ℒ(A) and ℒ(M) are retracts. Then, in a derivation of a cover, we can assume that only finite subsets occur on the right-hand side of the cover relation.