{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/86918"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/86918","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"The Injective Envelope as the Space of Extremal Functions","abstract":"In this thesis, we study the injective envelope of metric spaces by viewing it as the space of extremal functions as defined by Isbell. Extremal functions are also Kate˘tov functions, which satisfy two inequalities derived from the triangle inequality. One of these inequalities, along with a minimality requirement, is used to define the extremal functions. We compare the extremal functions to other classes of functions defined similarly using one of the two inequalities from the definition of Kate˘tov functions. We also consider separability of the space of extremal functions. We give a general method for generating uncountably many extremal functions from one extremal function satisfying certain inequalities on a sequence of ordered pairs. Then we prove non-separability of the space of extremal functions over some metric subspaces of finite dimensional real Banach spaces and some bounded metric spaces by constructing such an extremal function. Lastly, we discuss some connections with Melleray's work on separability of the space of Kate˘tov functions.","abstract_html":"In this thesis, we study the injective envelope of metric spaces by viewing it as the space of extremal functions as defined by Isbell. Extremal functions are also Kate˘tov functions, which satisfy two inequalities derived from the triangle inequality. One of these inequalities, along with a minimality requirement, is used to define the extremal functions. We compare the extremal functions to other classes of functions defined similarly using one of the two inequalities from the definition of Kate˘tov functions. We also consider separability of the space of extremal functions. We give a general method for generating uncountably many extremal functions from one extremal function satisfying certain inequalities on a sequence of ordered pairs. Then we prove non-separability of the space of extremal functions over some metric subspaces of finite dimensional real Banach spaces and some bounded metric spaces by constructing such an extremal function. Lastly, we discuss some connections with Melleray's work on separability of the space of Kate˘tov functions.","abstract_has_math":false,"creators":["Dong, Zhou"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Henson, C. Ward"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2015,"date_issued":"2015-09-28T15:20:10Z","date_published":"2015-09-28T15:20:10Z","updated_at":"2026-07-22T22:26:28Z","subjects":["Mathematics"],"languages":["eng"],"rights":[],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["(MiAaPQ)AAI3347297"],"render_values":[{"text":"(MiAaPQ)AAI3347297","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/2142/86918","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Henson, C. Ward"]},{"key":"dc:creator","label":"Author","values":["Dong, Zhou"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2015-09-28T15:20:10Z","10000-01-01","2008"]},{"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":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["eng"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["http://hdl.handle.net/2142/86918","(MiAaPQ)AAI3347297"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["In this thesis, we study the injective envelope of metric spaces by viewing it as the space of extremal functions as defined by Isbell. Extremal functions are also Kate˘tov functions, which satisfy two inequalities derived from the triangle inequality. One of these inequalities, along with a minimality requirement, is used to define the extremal functions. We compare the extremal functions to other classes of functions defined similarly using one of the two inequalities from the definition of Kate˘tov functions. We also consider separability of the space of extremal functions. We give a general method for generating uncountably many extremal functions from one extremal function satisfying certain inequalities on a sequence of ordered pairs. Then we prove non-separability of the space of extremal functions over some metric subspaces of finite dimensional real Banach spaces and some bounded metric spaces by constructing such an extremal function. Lastly, we discuss some connections with Melleray's work on separability of the space of Kate˘tov functions.","Made available in DSpace on 2015-09-28T15:20:10Z (GMT). No. of bitstreams: 2 license.txt: 4848 bytes, checksum: 96035ab3f5e1c23cc7138a224ce498bd (MD5) 3347297.pdf: 2791291 bytes, checksum: 22075ada0acd1bcfe68742778f8cc5ff (MD5) Previous issue date: 2008","Embargo set by: Seth Robbins for item 88199 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","88 p.","Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 2008."]},{"key":"dc:title","label":"Title","values":["The Injective Envelope as the Space of Extremal Functions"]}]}],"canonical_facts":{"dc:contributor":["Henson, C. Ward"],"dc:creator":["Dong, Zhou"],"dc:date":["2015-09-28T15:20:10Z","10000-01-01","2008"],"dc:description":["In this thesis, we study the injective envelope of metric spaces by viewing it as the space of extremal functions as defined by Isbell. Extremal functions are also Kate˘tov functions, which satisfy two inequalities derived from the triangle inequality. One of these inequalities, along with a minimality requirement, is used to define the extremal functions. We compare the extremal functions to other classes of functions defined similarly using one of the two inequalities from the definition of Kate˘tov functions. We also consider separability of the space of extremal functions. We give a general method for generating uncountably many extremal functions from one extremal function satisfying certain inequalities on a sequence of ordered pairs. Then we prove non-separability of the space of extremal functions over some metric subspaces of finite dimensional real Banach spaces and some bounded metric spaces by constructing such an extremal function. Lastly, we discuss some connections with Melleray's work on separability of the space of Kate˘tov functions.","Made available in DSpace on 2015-09-28T15:20:10Z (GMT). No. of bitstreams: 2 license.txt: 4848 bytes, checksum: 96035ab3f5e1c23cc7138a224ce498bd (MD5) 3347297.pdf: 2791291 bytes, checksum: 22075ada0acd1bcfe68742778f8cc5ff (MD5) Previous issue date: 2008","Embargo set by: Seth Robbins for item 88199 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","88 p.","Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 2008."],"dc:identifier":["http://hdl.handle.net/2142/86918","(MiAaPQ)AAI3347297"],"dc:language":["eng"],"dc:subject":["Mathematics"],"dc:title":["The Injective Envelope as the Space of Extremal Functions"],"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":"2026-07-22T22:26:28Z"}