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Title:  NearAtomic Spaces 
Author(s):  Evans, Dennis Neal 
Doctoral Committee Chair(s):  Peck, T., 
Department / Program:  Mathematics 
Discipline:  Mathematics 
Degree Granting Institution:  University of Illinois at UrbanaChampaign 
Degree:  Ph.D. 
Genre:  Dissertation 
Subject(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 trivialdual 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 Fspaces 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 Fspace ($X,\Vert\cdot\Vert$) 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 $$E\sb{(\sum\sbsp{i}{k1}n\sb{i})+1}\cup\ E \sb{(\sum\sbsp{i}{k1}n\sb{i})+1}\cup\ \cdots\ E\sb{\sum\sbsp{i}{k}n\sb{i}}.$$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 Fspaces ($X,\Vert\cdot\Vert$), 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 quasibasis of V. We also show that V is a needlepoint space. 
Issue Date:  1993 
Type:  Text 
Description:  145 p. Thesis (Ph.D.)University of Illinois at UrbanaChampaign, 1993. 
URI:  http://hdl.handle.net/2142/72537 
Other Identifier(s):  (UMI)AAI9314864 
Date Available in IDEALS:  20141217 
Date Deposited:  1993 
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Dissertations and Theses  Mathematics

Graduate Dissertations and Theses at Illinois
Graduate Theses and Dissertations at Illinois