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...

This document contains two examples of the use of distributive lattices as spaces in commutative algebra. The first example is a simple proof of Forster’s Theorem about the number of generators over a ring of finite Krull dimension. The second example is the beginning of the theory of Prüfer Domain, which has to be thought of as a non Noetherian version of the theory of Dedekind

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ABSTRACT. We investigate the connection between the spatiality of locale products and the earlier studies of the author on the locally fine coreflection of the products of uniform spaces. After giving a historical introduction and indicating the connection between spatiality and the locally fine construction, we indicate how the earlier results directly solve the first of the two open problems announced in the thesis of T. Plewe. Finally, we establish a general isomorphism between the covering monoids of the localic product of topological (completely regular) spaces and the locally fine coreflection of the corresponding product of (fine) uniform spaces. Additionally, paper relates the recent studies on formal topology and uniform spaces by showing how the transitivity of covering relations corresponds to the locally fine construction.