Near-Atomic Spaces

Evans, Dennis Neal

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https://hdl.handle.net/2142/72537 Copy

Description

Title

Near-Atomic Spaces

Author(s)

Evans, Dennis Neal

Issue Date

1993

Doctoral Committee Chair(s)

Peck, T.,

Department of Study

Mathematics

Discipline

Mathematics

Degree Granting Institution

University of Illinois at Urbana-Champaign

Degree Name

Ph.D.

Degree Level

Dissertation

Keyword(s)

Mathematics

Abstract

• The examples discussed in this paper are related to the atomic space problem: Is there an infinite dimensional space with no proper closed infinite dimensional subspace? This question is equivalent to one first posed by A. Pelczynski; namely, does every infinite dimensional metric linear space have an infinite dimensional subspace with a nonzero continuous linear functional?
• An atomic space, if one were to exist, would represent a delightful anomaly in the theory of metric linear spaces. If an infinite dimensional metric linear space is endowed with a basis then it is necessarily nonatomic. Within the class of nonlocally convex spaces, each of the classical examples is nonatomic. For instance, $L\sb0\lbrack 0,1\rbrack$ is a common example of a trivial-dual space. It is quickly seen that the set of measurable functions supported over (0, 1/2) is a proper closed infinite dimensional subspace of $L\sb0\lbrack 0,1\rbrack$.
• In this paper, we develop a class of F-spaces with pathologies similar to those of the hypothetical atomic space, and examine the properties of a typical representative ($V,\Vert\cdot\Vert$) of this class.
• Theorem. For every sequence of natural numbers $\langle s\sb{k}\rangle\sbsp{k=1}{\infty}$, there is a F-space ($X,\Vert\cdot\Vert$) with a set $\{E\sb{k}\}\sbsp{k=1}{\infty}$ of (independent) finite dimensional subspaces ($\dim E\sb{k}=s\sb{k}$) and the property: Let $\langle n\sb{k}\rangle\sbsp{k=1}{\infty}$ be any bounded sequence of natural numbers, and, for each natural number k, define $F\sb{k}$ to be the smallest subspace of X containing $$E\sb{(\sum\sbsp{i}{k-1}n\sb{i})+1}\cup\ E \sb{(\sum\sbsp{i}{k-1}n\sb{i})+1}\cup\ \cdots\ E\sb{\sum\sbsp{i}{k}n\sb{i}}.$$If $\langle x\sb{k}\rangle\sbsp{k=1}{\infty}$ is a sequence in X such that $x\sb{k}$ is in $F\sb{k}$ for each natural number k, and all but finitely many of the $x\sb{k}$'s are nonzero, then ($x\sb{k}\rbrack\sbsp{k=1}{\infty}$ (the linear span of the set $\{x\sb{k}\}\sbsp{k=1}{\infty}$) is dense in X.
• For each of these F-spaces ($X,\Vert\cdot\Vert$), X is taken to be the set of sequences of real numbers which are eventually zero. We show that although V does not contain a basis, the set of coordinate vectors $\{e\sb{n}\}\sbsp{n=1}{\infty}$ serves as a quasi-basis of V. We also show that V is a needlepoint space.

Graduation Semester

1993

Type of Resource

text

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http://hdl.handle.net/2142/72537

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