The Hahn-Banach Theorem in Type Theory

We give the basic de nitions for pointfree functional analysis and present constructive proofs of the Alaoglu and Hahn Banach theorems in the set ting of formal topology Introduction We present the basic concepts and de nitions 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 and by Vermeulen A constructive proof based on points was given by Bishop In the formulation of his proof the norm of the linear functional is pre served 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 de ne the objects under analysis as formal points of a suitable formal space After this has been accom plished 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 functionals of norm 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 says that the restriction mapping from the linear functionals on A of norm to those on M is surjective In terms of covers conceived as deductive systems it becomes a conservativity statement cf Mulvey and Pelletier Whenever a is an As explained by Hochstad the main idea in the usual proof of what is called the Hahn Banach theorem is due to Helly Since this is also the key idea in our derivation we here rename the theorem in this way J Cederquist T Coquand and S Negri 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 nd 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 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 nite subsets occur on the right hand side of the cover relation A global proof transformation makes it possible to change a derivation in L A into a derivation in L M since only a nite dimensional extension of the space M has to be taken into account In consequence as in the case for the classical proof for the one dimensional extension no use of Zorn s lemma is needed There exist already two pointfree proofs of Hahn Banach s theorem The proof of Mulvey and Pelletier shows Hahn Banach s theorem in any Grothendieck topos However the argument relies on Barr s theorem for which no constructive justi cation has been given so far The proof of Vermeulen is done in the framework of topos theory with a natural number object and thus a priori relies on the use of impredicative quanti cation Here as elsewhere cf Cederquist and Negri Coquand Ne gri and Valentini the use of formal topology allows for elementary and constructive proofs of pointfree formulations of classical results The two main contributions of this paper are the following Our proof of the pointfree version of the Hahn Banach theorem following rather closely the original proof by Helly This proof can actually be expressed in Martin L f s Type Theory cf Martin L f In fact on the basis of our proof the rst author has done a formalisation of the Hahn Banach theorem in an implementation of the intensional version of Type Theory with one universe and nitary inductive de nitions see Cederquist Preliminaries We recall here the de nition of formal topology introduced by Per Martin L f and Giovanni Sambin see Sambin We remark that in contrast to the de nition given by Sambin and without any substantial di erence in the devel opment of the theory we do not require the base monoid to have a unit Nor do we have the positivity predicate used by Sambin De nition Formal topology A formal topology over a set S is a struc ture hS i where hS i is a commutative monoid is a relation called cover between elements and subsets of S such that for any a b S and U V S the following conditions hold The Hahn Banach Theorem in Type Theory