Aspects of the Theory of Normed Spaces

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.</p

Analysis of methods for extraction of programs from non-constructive proofs

The present thesis compares two computational interpretations of non-constructive proofs: refined A-translation and Gödel's functional "Dialectica" interpretation. The behaviour of the extraction methods is evaluated in the light of several case studies, where the resulting programs are analysed and compared. It is argued that the two interpretations correspond to specific backtracking implementations and that programs obtained via the refined A-translation tend to be simpler, faster and more readable than programs obtained via Gödel's interpretation. Three layers of optimisation are suggested in order to produce faster and more readable programs. First, it is shown that syntactic repetition of subterms can be reduced by using let-constructions instead of meta substitutions abd thus obtaining a near linear size bound of extracted terms. The second improvement allows declaring syntactically computational parts of the proof as irrelevant and that this can be used to remove redundant parameters, possibly improving the efficiency of the program. Finally, a special case of induction is identified, for which a more efficient recursive extracted term can be defined. It is shown the outcome of case distinctions can be memoised, which can result in exponential improvement of the average time complexity of the extracted program