Every continuous convex function defined on a separable Banach space is Gateaux differentiable on a dense G^ subset of the space E [Mazur]. Suppose we are given a sequence (xn) that Is dense in E. Can we always find a Gateaux differentiable point x such that x = z^=^anxn.for some sequence (an) with infinitely many non-zero terms so that Ση∞=1||anxn|| < co ? According to this paper, such points are called of "simple type," and shown to be dense in E. Mazur's theorem follows directly from the result and Rybakov's theorem (A countably additive vector measure F: E -* X on a cr-field is absolutely continuous with respect to |x*F] for some x* e Xs) can be shown without deep measure theoretic Involvement.
| Identifer | oai:union.ndltd.org:unt.edu/info:ark/67531/metadc331018 |
| Date | 12 1900 |
| Creators | Oh, Seung Jae |
| Contributors | Bilyeu, Russell Gene, Mauldin, R. Daniel, Connor, Frank |
| Publisher | North Texas State University |
| Source Sets | University of North Texas |
| Language | English |
| Detected Language | English |
| Type | Thesis or Dissertation |
| Format | iii, 24 leaves, Text |
| Rights | Public, Oh, Seung Jae., Copyright, Copyright is held by the author, unless otherwise noted. All rights reserved. |
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