Coanalytic subsets of some well known Polish spaces are investigated. A natural norm (rank function) on each subset is defined and studied by using well-founded trees and transfinite induction as the main tools. The norm provides a natural measure of the complexity of the elements in each subset. It also provides a \"Rank Argument\" of the non-Borelness of the subset.

\r\n\r\nThe work is divided into four chapters. In Chapter 1 nowhere differentiable continuous functions and Besicovitch functions are studied. Chapter 2 deals with functions with everywhere divergent Fourier series, and everywhere divergent trigonometric series with coefficients that tend to zero. Compact Jordan sets (i.e., sets without cavities) and compact simply-connected sets in the plane are investigated in Chapter 3. Chapter 4 is a miscellany of results extending earlier work of M. Ajtai, A. Kechris and H. Woodin on differentiable functions and continuous functions with everywhere convergent Fourier series.

", "doi": "10.7907/8pdn-xf41", "publication_date": "1986", "thesis_type": "phd", "thesis_year": "1986" }, { "id": "thesis:10092", "collection": "thesis", "collection_id": "10092", "cite_using_url": "https://resolver.caltech.edu/CaltechTHESIS:03092017-112801788", "primary_object_url": { "basename": "WIID_FGJ_1982.pdf", "content": "final", "filesize": 27387023, "license": "other", "mime_type": "application/pdf", "url": "/10092/1/WIID_FGJ_1982.pdf", "version": "v3.0.0" }, "type": "thesis", "title": "Aspects of the Theory of Normed Spaces", "author": [ { "family_name": "Wiid", "given_name": "Frans Gerhardus Johannes", "clpid": "Wiid-Frans-Gerhardus-Johannes" } ], "thesis_advisor": [ { "family_name": "Luxemburg", "given_name": "W. A. J.", "clpid": "Luxemburg-W-A-J" } ], "thesis_committee": [ { "family_name": "Luxemburg", "given_name": "W. A. J.", "clpid": "Luxemburg-W-A-J" }, { "family_name": "Woodin", "given_name": "W. Hugh", "clpid": "Woodin-W-H" }, { "family_name": "De Prima", "given_name": "Charles R.", "clpid": "De-Prima-C-R" }, { "family_name": "Fuller", "given_name": "F. Brock", "clpid": "Fuller-F-Brock" } ], "local_group": [ { "literal": "div_pma" } ], "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.

\r\n\r\nThe 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\r\n2-category of Topos-valued normed spaces, and the study of the elementary properties of these adjoints.

\r\n\r\nThe 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.

\r\n", "doi": "10.7907/rytj-9p73", "publication_date": "1982", "thesis_type": "phd", "thesis_year": "1982" } ]