Constructive and intuitionistic integration theory and functional analysis

B.A.W. Spitters

    Research output: ThesisPhd Thesis 4 Research NOT TU/e / Graduation NOT TU/e)

    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.
    Original languageEnglish
    QualificationDoctor of Philosophy
    Awarding Institution
    • Radboud University Nijmegen
    Supervisors/Advisors
    • Rooij, van, A.C.M., Promotor, External person
    • Barendregt, H.P., Promotor, External person
    • Veldman, W.H.M., Copromotor, External person
    Award date27 Feb 2003
    Place of PublicationNijmegen
    Publisher
    Print ISBNs90-9016281-X
    Publication statusPublished - 2002

    Fingerprint

    Functional Analysis
    Representation Theorem
    Hilbert space
    Constructive Mathematics
    Spectral Theorem
    Weyl's Theorem
    Measurable space
    Almost Periodic Functions
    Normal Operator
    Algebra
    Spectral Measure
    Ergodic Theorem
    Approximation Theorem
    Unbounded Operators
    Compact Group
    Von Neumann Algebra
    Substitute
    Compact Set
    Programming
    Physics

    Cite this

    Spitters, B. A. W. (2002). Constructive and intuitionistic integration theory and functional analysis. Nijmegen: Radboud Universiteit Nijmegen.
    Spitters, B.A.W.. / Constructive and intuitionistic integration theory and functional analysis. Nijmegen : Radboud Universiteit Nijmegen, 2002. 127 p.
    @phdthesis{766ece7b5b05415abb85bf6a28106bbd,
    title = "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.",
    author = "B.A.W. Spitters",
    year = "2002",
    language = "English",
    isbn = "90-9016281-X",
    publisher = "Radboud Universiteit Nijmegen",
    school = "Radboud University Nijmegen",

    }

    Spitters, BAW 2002, 'Constructive and intuitionistic integration theory and functional analysis', Doctor of Philosophy, Radboud University Nijmegen, Nijmegen.

    Constructive and intuitionistic integration theory and functional analysis. / Spitters, B.A.W.

    Nijmegen : Radboud Universiteit Nijmegen, 2002. 127 p.

    Research output: ThesisPhd Thesis 4 Research NOT TU/e / Graduation NOT TU/e)

    TY - THES

    T1 - Constructive and intuitionistic integration theory and functional analysis

    AU - Spitters, B.A.W.

    PY - 2002

    Y1 - 2002

    N2 - 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.

    AB - 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.

    UR - http://repository.ubn.ru.nl/handle/2066/19238

    M3 - Phd Thesis 4 Research NOT TU/e / Graduation NOT TU/e)

    SN - 90-9016281-X

    PB - Radboud Universiteit Nijmegen

    CY - Nijmegen

    ER -

    Spitters BAW. Constructive and intuitionistic integration theory and functional analysis. Nijmegen: Radboud Universiteit Nijmegen, 2002. 127 p.