A constructive view on ergodic theorems

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Constructive Results on Operator Algebras

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Constructive algebraic integration theory without choice

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Formal Topology and Constructive Mathematics: the Gelfand and Stone-Yosida Representation Theorems

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Constructive and intuitionistic integration theory and functional analysis

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

Locating the range of an operator with an adjoint

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A constructive converse of the mean value theorem

Consider the following converse of the Mean Value Theorem. Let f be a differentiable function on [a, b]. If c e (a, b), then there are a and ß in [a, b] such that (f(ß) - f(a))/(ß - a) = f'(c). Assuming some weak conditions to be mentioned in Section 3, Tong and Braza [3] were able to prove this statement. Unfortunately their proof does not provide a method to compute a and ß. We give a constructive proof

A constructive proof of the Peter-Weyl theorem

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Computable sets : located and overt locales

What is a computable set? One may call a bounded subset of the plane computable if it can be drawn at any resolution on a computer screen. Using the constructive approach to computability one naturally considers totally bounded subsets of the plane. We connect this notion with notions introduced in other frameworks. A subset of a totally bounded set is again totally bounded iff it is located. Locatedness is one of the fundamental notions in constructive mathematics. The existence of a positivity predicate on a locale, i.e. the locale being overt, or open, has proved to be fundamental in locale theory in a constructive, or topos theoretic, context. We show that the two notions are intimately connected. We propose a definition of located closed sublocale motivated by locatedness of subsets of metric spaces. A closed sublocale of a compact regular locale is located iff it is overt. Moreover, a closed subset of a complete metric space is Bishop compact — that is, totally bounded and complete — iff its localic completion is compact overt. For Baire space metric locatedness corresponds to having a decidable positivity predicate. Finally, we show that the points of the Vietoris locale of a compact regular locale are precisely its compact overt sublocales. We work constructively, predicatively and avoid the use of the axiom of countable choice. Consequently, all are results are valid in any predicative topos

Developing the algebraic hierarchy with type classes in Coq

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