Abstract

We give the basic deønitions for pointfree functional analysis and present constructive proofs of the Alaoglu and Hahn-Banach theorems in the setting of formal topology. 1 Introduction We present the basic concepts and deønitions 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 1 theorems. Earlier pointfree formulations of the Hahn-Banach theorem, in a topostheoretic 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 deøne the objects under analysis as formal points of a suitable formal space. After this has...

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