A space of spaces

In elementary functional analysis, one is taught that all normed vector spaces of given finite dimension over the reals or the complex numbers are isomorphic. One therefore could conclude that the theory of finite-dimensional vector spaces is trivial and not worth investigating. In this bachelor thesis, I will try to convince the reader that this is not the case. For instance, when investigating general Banach spaces one comes upon the field of local theory. This research area studies finite-dimensional normed spaces and the relation between a Banach space and its finite-dimensional subspaces. We will delve into a small part of this theory by introducing the BanachMazur distance, a fundamental tool when studying finite-dimensional normed spaces. Methods developed in the study of this distance have helped answer several longstanding questions about Banach spaces, and are used in other areas such as operator theory and harmonic analysis [Tomczak-Jaegermann 1989, p. x,xi]. The Banach...Show moreIn elementary functional analysis, one is taught that all normed vector spaces of given finite dimension over the reals or the complex numbers are isomorphic. One therefore could conclude that the theory of finite-dimensional vector spaces is trivial and not worth investigating. In this bachelor thesis, I will try to convince the reader that this is not the case. For instance, when investigating general Banach spaces one comes upon the field of local theory. This research area studies finite-dimensional normed spaces and the relation between a Banach space and its finite-dimensional subspaces. We will delve into a small part of this theory by introducing the BanachMazur distance, a fundamental tool when studying finite-dimensional normed spaces. Methods developed in the study of this distance have helped answer several longstanding questions about Banach spaces, and are used in other areas such as operator theory and harmonic analysis [Tomczak-Jaegermann 1989, p. x,xi]. The Banach-Mazur distance defines a metric on a set of equivalence classes of normed spaces, thereby defining the ’space of spaces’ from the title of this thesis. We can then ask ourselves several questions about this space, amongst which whether the space is bounded and compact. The first questions leads to the theorem of John (1948), which gives an upper bound for the diameter of this space. The problem of determining the quality of this upper bound was left open for some time, until it was solved in 1981 by Gluskin. His proof uses an interesting measure-theoretic approach which has since then been used in several other results. This theorem is the main piece of this thesis, and we will work out in more detail his short but very technical proof [Gluskin 1981]. In the paragraph following this introduction, we introduce the Banach-Mazur distance as a way of measuring how different two norms on isomorphic vector spaces are. We explore some of the properties of this distance. Using this distance, we can define a metric on the set of equivalence classes of n-dimensional (n ∈ N) normed spaces over the reals, and in the third section we go on to explore some of the properties of this metric space. In the fourth paragraph, we introduce the theorem of Gluskin to which we alluded previously, and go on to prove it, using the results from two appendices. We conclude with a paragraph stating some additional results using the same technique as Gluskin’s proofShow less

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A space of spaces