Constructive and intuitionistic integration theory and functional analysis

Abstract

There are ample reasons to develop mathematics constructively, for instance because one is interested in the foundations of mathematics or in programming on a highly abstract level. But is it possible to do advanced mathematics constructively? Bishop and his followers showed that large parts of abstract analysis and algebra can be developed constructively. But nevertheless the discussion about the applicability of constructive mathematics to mathematical physics started again. This discussion gave rise to a number of precise mathematical problems. Some of these problems have already been solved, others will be answered in this thesis. These problems all belong to functional analysis, part of which is developed constructively in this thesis. To be more precise, the thesis contains a constructive substitute for the ergodic theorem, a constructive Peter-Weyl theorem for representations of compact groups, approximation theorems for almost periodic functions and a spectral theorem for unbounded normal operators on Hilbert spaces. Moreover, parts of the theory of algebras of operators are developed constructively: two representation theorems are proved, one for finite dimensional and one for Abelian von Neumann algebras. A simple proof is given that the spectral measure is independent of the choice of the basis of the underlying Hilbert space. Finally, a representation theorem for normal functionals is proved. As a tool which is interesting in its own right parts of integration theory are redeveloped, using a metric related to convergence in measure. It is proved that many measures are regular, that is, integrable sets can be approximated by compact sets. This result is used to extend some important intuitionistic theorems. Finally, a representation theorem for Chan's measurable spaces is given, making it possible to understand these spaces intuitionistically