The structure and geometry of subspaces of a given Banach space is among the most fundamental questions in Functional Analysis. In 1961, Kadec and Pelczyński pioneered a field of study by analyzing the structures of subspaces and basic sequences in L_p[0,1] under a naturally occurring restriction of p, 2 < p <\infty. They proved that any infinite-dimensional subspace X\subset L_p[0,1] for 2<p<\infty must either be isomorphic to l_2 and complemented in L_p or must contain a complemented subspace which is isomorphic to l_p. Many works since have studied the relationships between the sides of this dichotomy, chiefly by weakening hypotheses on side of the equation to gain stronger assumptions on the other. In this way, Johnson and Odell were able to show in 1974 that if X contains no further subspace which is isomorphic to l_2, then it must embed into l_p. Kalton and Werner further strengthened this result in 1993 by showing that such an embedding must be almost isometric.
We start by analyzing the tools and definitions originally introduced in 1961 and define a natural extension to these methods. By analyzing this extension, we provide a constructive and streamlined reproving of Kalton and Werner's theorem:
Let X be an infinite dimensional subspace of L_p[0,1] for 2<p<\infty. Then, either X contains a subspace which is isomorphic to l_2, or for every \varepsilon>\ 0, X embeds into l_p with constant 1 + \varepsilon.
| Identifer | oai:union.ndltd.org:unt.edu/info:ark/67531/metadc1833434 |
| Date | 08 1900 |
| Creators | James, Christopher W |
| Contributors | Sari, Bunyamin, Jackson, Stephen, Urbanski, Mariusz |
| Publisher | University of North Texas |
| Source Sets | University of North Texas |
| Language | English |
| Detected Language | English |
| Type | Thesis or Dissertation |
| Format | iv, 47 pages, Text |
| Rights | Public, James, Christopher W, Copyright, Copyright is held by the author, unless otherwise noted. All rights Reserved. |
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