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Aspects of the Theory of Normed Spaces

Citation

Wiid, Frans Gerhardus Johannes (1982) Aspects of the Theory of Normed Spaces. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/rytj-9p73. https://resolver.caltech.edu/CaltechTHESIS:03092017-112801788

Abstract

The dissertation will be divided into two parts. The first part will, in essence, be a study of weak compactness in a variety of families of normed spaces. Included in this study will be general characterizations of weak compactness in spaces of vector measures and tensor products that contain all known results of this nature as special cases (in particular, we do not need to restrict attention to only those range spaces with strong geometric properties such as, for example, the Radon-Nikodym property). The methods of Nonstandard Analysis constitute a fundamental tool in these investigations.

The second part of the dissertation will contain a discussion and a study of Model theoretic aspects of categories of normed spaces. We will introduce multi-sorted formal languages that enable us to view various subcategories of the category of normed spaces as being equivalent to categories of set-valued models of coherent theories in these languages. We see, in particular, that the category of real normed spaces is equivalent to the category of set-valued models of a lim-theory, and that, for instance, the category of L-spaces is equivalent to the category of set-valued models of a coherent extension of this lim-theory. These considerations allow for proofs of existence of 2-adjoints to inclusion functors from some 2-categories into the 2-category of Topos-valued normed spaces, and the study of the elementary properties of these adjoints.

The coherent theory of Hilbert spaces gives rise to interesting spatial Toposes when the appropriate "adjoint functor theorems" are proved. The sites of these toposes are spectral spaces (in the sense of Algebraic geometry) with interesting cohomological properties.

Item Type:Thesis (Dissertation (Ph.D.))
Subject Keywords:Mathematics
Degree Grantor:California Institute of Technology
Division:Physics, Mathematics and Astronomy
Major Option:Mathematics
Thesis Availability:Public (worldwide access)
Research Advisor(s):
  • Luxemburg, W. A. J.
Thesis Committee:
  • Luxemburg, W. A. J. (chair)
  • Woodin, W. Hugh
  • DePrima, Charles R.
  • Fuller, F. Brock
Defense Date:21 May 1982
Funders:
Funding AgencyGrant Number
Council for Scientific and Industrial ResearchUNSPECIFIED
CaltechUNSPECIFIED
Record Number:CaltechTHESIS:03092017-112801788
Persistent URL:https://resolver.caltech.edu/CaltechTHESIS:03092017-112801788
DOI:10.7907/rytj-9p73
Default Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:10092
Collection:CaltechTHESIS
Deposited By: Bianca Rios
Deposited On:09 Mar 2017 23:26
Last Modified:16 Apr 2021 22:19

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