{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/72537"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/72537","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Near-Atomic Spaces","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?","abstract_html":"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?","abstract_has_math":false,"creators":["Evans, Dennis Neal"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Peck, T.,"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2014,"date_issued":"2014-12-17T23:17:46Z","date_published":"2014-12-17T23:17:46Z","updated_at":"2026-07-22T22:26:07Z","subjects":["Mathematics"],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["(UMI)AAI9314864"],"render_values":[{"text":"(UMI)AAI9314864","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/2142/72537","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Peck, T.,"]},{"key":"dc:creator","label":"Author","values":["Evans, Dennis Neal"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2014-12-17T23:17:46Z","10000-01-01","1993"]},{"key":"dc:type","label":"Dc Type","values":["text"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph.D."]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["University of Illinois at Urbana-Champaign"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Mathematics"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["http://hdl.handle.net/2142/72537","(UMI)AAI9314864"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["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.","Made available in DSpace on 2014-12-17T23:17:46Z (GMT). No. of bitstreams: 1 9314864.pdf: 5172562 bytes, checksum: f2ea15120ebb3d716a76fd8bf32233e7 (MD5) Previous issue date: 1993","Embargo set by: Seth Robbins for item 72705 Lift date: Forever Reason: Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs","Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs","U of I Only","145 p.","Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1993."]},{"key":"dc:title","label":"Title","values":["Near-Atomic Spaces"]}]}],"canonical_facts":{"dc:contributor":["Peck, T.,"],"dc:creator":["Evans, Dennis Neal"],"dc:date":["2014-12-17T23:17:46Z","10000-01-01","1993"],"dc:description":["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.","Made available in DSpace on 2014-12-17T23:17:46Z (GMT). No. of bitstreams: 1 9314864.pdf: 5172562 bytes, checksum: f2ea15120ebb3d716a76fd8bf32233e7 (MD5) Previous issue date: 1993","Embargo set by: Seth Robbins for item 72705 Lift date: Forever Reason: Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs","Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs","U of I Only","145 p.","Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1993."],"dc:identifier":["http://hdl.handle.net/2142/72537","(UMI)AAI9314864"],"dc:subject":["Mathematics"],"dc:title":["Near-Atomic Spaces"],"dc:type":["text"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Ph.D."],"thesis:institution_name":["University of Illinois at Urbana-Champaign"]},"updated_at":"2026-07-22T22:26:07Z"}