Date of Award
Doctor of Philosophy
Theory of Banach spaces as a branch of Functional Analysis was founded at the beginning of the 20th century and since then it has been extensively developed and applied. Banach lattices, in particular Banach function and sequence spaces, like classical Lebesgue spaces or Orlicz, Lorentz, Musielak-Orlicz spaces, are of special interest and importance for applications. In this dissertation we study Cesàro function and sequence spaces which, unlike many other classical spaces, are not rearrangement invariant. This makes them a very interesting object to explore with possible applications to some functional analysis problems. Cesàro function and sequence spaces appeared for the first time in 1968 in the Dutch Mathematical Society Journal as a problem to find a representation of their dual space. This problem, in case of sequences, was solved in 1974 by Jagers. Since then, Cesàro sequence spaces have gain an attention among Banach space theory specialists. For example, in the nineties several geometric properties of them were studied. During the last decade, more general spaces, Cesàro-Orlicz sequence spaces, have been explored as well. Cesàro function spaces attracted a wider attention only since 2008 when Astashkin and Maligranda published series of interesting papers on their geometrical and topological properties. In this dissertation we study geometric properties of Cesàro function spaces with general weight. We find an isometric description of the dual of Cesàro function space. This description involves a new concept of essential Psi-concave majorant of a measurable function which we define and study. We show, among others, that every non empty relatively open subset (and hence every slice) of the unit ball of Cesàro function space has diameter 2. In particular these spaces do not have the Radon-Nikodym property. Also, they are strictly convex Banach spaces while the unit sphere does not have strongly extreme points. This shows rather unexpected differences between Cesàro function and sequence spaces since the latter are known to have the Radon-Nikodym property and to be locally uniformly rotund.We also explore Cesàro-Orlicz sequence spaces. We show that they are not B-convex and investigate under what conditions there is an isometric or isomorphic copy of ∞ in these spaces.
dissertation or thesis originally submitted to the local University of Memphis Electronic Theses & dissertation (ETD) Repository.
Kubiak, Damian Marcin, "Geometric Properties of Cesro Function and Sequence Spaces" (2012). Electronic Theses and Dissertations. 459.