{"id":"https://openalex.org/W951606502","doi":"https://doi.org/10.2478/forma-2014-0007","title":"Dual Spaces and Hahn-Banach Theorem","display_name":"Dual Spaces and Hahn-Banach Theorem","publication_year":2014,"publication_date":"2014-03-30","ids":{"openalex":"https://openalex.org/W951606502","doi":"https://doi.org/10.2478/forma-2014-0007","mag":"951606502"},"language":"en","primary_location":{"id":"doi:10.2478/forma-2014-0007","is_oa":true,"landing_page_url":"https://doi.org/10.2478/forma-2014-0007","pdf_url":"https://content.sciendo.com/downloadpdf/journals/forma/22/1/article-p69.pdf","source":{"id":"https://openalex.org/S179241345","display_name":"Formalized Mathematics","issn_l":"1426-2630","issn":["1426-2630","1898-9934"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320322","host_organization_name":"De Gruyter Open","host_organization_lineage":["https://openalex.org/P4310320322","https://openalex.org/P4310313990"],"host_organization_lineage_names":["De Gruyter Open","De Gruyter"],"type":"journal"},"license":"cc-by-sa","license_id":"https://openalex.org/licenses/cc-by-sa","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Formalized Mathematics","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"diamond","oa_url":"https://content.sciendo.com/downloadpdf/journals/forma/22/1/article-p69.pdf","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5057236933","display_name":"Keiko Narita","orcid":null},"institutions":[{"id":"https://openalex.org/I137975476","display_name":"Shinshu University","ror":"https://ror.org/0244rem06","country_code":"JP","type":"education","lineage":["https://openalex.org/I137975476"]},{"id":"https://openalex.org/I4210113483","display_name":"National Institute of Technology, Gifu College","ror":"https://ror.org/025b2hx26","country_code":"JP","type":"education","lineage":["https://openalex.org/I4210113483","https://openalex.org/I4210120810"]},{"id":"https://openalex.org/I4210149837","display_name":"Hirosaki National Hospital","ror":"https://ror.org/03qrrp624","country_code":"JP","type":"healthcare","lineage":["https://openalex.org/I4210137409","https://openalex.org/I4210149837"]},{"id":"https://openalex.org/I4210162928","display_name":"\u014ctani University","ror":"https://ror.org/05b7rex33","country_code":"JP","type":"education","lineage":["https://openalex.org/I4210162928"]}],"countries":["JP"],"is_corresponding":false,"raw_author_name":"Keiko Narita","raw_affiliation_strings":["Hirosaki-city Aomori, Japan","Yasunari Shidama Shinshu University Nagano, Japan","Gifu National College of Technology Gifu, Japan"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Hirosaki-city Aomori, Japan","institution_ids":["https://openalex.org/I4210149837"]},{"raw_affiliation_string":"Yasunari Shidama Shinshu University Nagano, Japan","institution_ids":["https://openalex.org/I137975476","https://openalex.org/I4210162928"]},{"raw_affiliation_string":"Gifu National College of Technology Gifu, Japan","institution_ids":["https://openalex.org/I4210113483"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5015450295","display_name":"Noboru Endou","orcid":"https://orcid.org/0000-0002-5922-2332"},"institutions":[{"id":"https://openalex.org/I137975476","display_name":"Shinshu University","ror":"https://ror.org/0244rem06","country_code":"JP","type":"education","lineage":["https://openalex.org/I137975476"]},{"id":"https://openalex.org/I4210113483","display_name":"National Institute of Technology, Gifu College","ror":"https://ror.org/025b2hx26","country_code":"JP","type":"education","lineage":["https://openalex.org/I4210113483","https://openalex.org/I4210120810"]},{"id":"https://openalex.org/I4210149837","display_name":"Hirosaki National Hospital","ror":"https://ror.org/03qrrp624","country_code":"JP","type":"healthcare","lineage":["https://openalex.org/I4210137409","https://openalex.org/I4210149837"]},{"id":"https://openalex.org/I4210162928","display_name":"\u014ctani University","ror":"https://ror.org/05b7rex33","country_code":"JP","type":"education","lineage":["https://openalex.org/I4210162928"]}],"countries":["JP"],"is_corresponding":false,"raw_author_name":"Noboru Endou","raw_affiliation_strings":["Gifu National College of Technology Gifu, Japan","Yasunari Shidama Shinshu University Nagano, Japan","Hirosaki-city Aomori, Japan"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Gifu National College of Technology Gifu, Japan","institution_ids":["https://openalex.org/I4210113483"]},{"raw_affiliation_string":"Yasunari Shidama Shinshu University Nagano, Japan","institution_ids":["https://openalex.org/I137975476","https://openalex.org/I4210162928"]},{"raw_affiliation_string":"Hirosaki-city Aomori, Japan","institution_ids":["https://openalex.org/I4210149837"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5055549852","display_name":"Yasunari Shidama","orcid":"https://orcid.org/0000-0001-5634-0838"},"institutions":[{"id":"https://openalex.org/I137975476","display_name":"Shinshu University","ror":"https://ror.org/0244rem06","country_code":"JP","type":"education","lineage":["https://openalex.org/I137975476"]}],"countries":["JP"],"is_corresponding":false,"raw_author_name":"Yasunari Shidama","raw_affiliation_strings":["Shinshu University Nagano, Japan"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Shinshu University Nagano, Japan","institution_ids":["https://openalex.org/I137975476"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":3,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":2.0162,"has_fulltext":true,"cited_by_count":7,"citation_normalized_percentile":{"value":0.86193753,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":97},"biblio":{"volume":"22","issue":"1","first_page":"69","last_page":"77"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12062","display_name":"Approximation Theory and Sequence Spaces","score":0.9919999837875366,"subfield":{"id":"https://openalex.org/subfields/2613","display_name":"Statistics and Probability"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T12062","display_name":"Approximation Theory and Sequence Spaces","score":0.9919999837875366,"subfield":{"id":"https://openalex.org/subfields/2613","display_name":"Statistics and Probability"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11777","display_name":"Fixed Point Theorems Analysis","score":0.9858999848365784,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11545","display_name":"Advanced Banach Space Theory","score":0.9746999740600586,"subfield":{"id":"https://openalex.org/subfields/2610","display_name":"Mathematical Physics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.8392942547798157},{"id":"https://openalex.org/keywords/interpolation-space","display_name":"Interpolation space","score":0.6889162063598633},{"id":"https://openalex.org/keywords/reflexive-space","display_name":"Reflexive space","score":0.6872791647911072},{"id":"https://openalex.org/keywords/lp-space","display_name":"Lp space","score":0.6806066632270813},{"id":"https://openalex.org/keywords/fr\u00e9chet-space","display_name":"Fr\u00e9chet space","score":0.6682255864143372},{"id":"https://openalex.org/keywords/eberlein\u2013\u0161mulian-theorem","display_name":"Eberlein\u2013\u0160mulian theorem","score":0.6407216787338257},{"id":"https://openalex.org/keywords/dual-norm","display_name":"Dual norm","score":0.6362659335136414},{"id":"https://openalex.org/keywords/open-mapping-theorem","display_name":"Open mapping theorem (functional analysis)","score":0.5871356129646301},{"id":"https://openalex.org/keywords/banach-space","display_name":"Banach space","score":0.5859484672546387},{"id":"https://openalex.org/keywords/birnbaum\u2013orlicz-space","display_name":"Birnbaum\u2013Orlicz space","score":0.5774690508842468},{"id":"https://openalex.org/keywords/dual","display_name":"Dual (grammatical number)","score":0.5741879343986511},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.5355809330940247},{"id":"https://openalex.org/keywords/topological-tensor-product","display_name":"Topological tensor product","score":0.4879915118217468},{"id":"https://openalex.org/keywords/banach-manifold","display_name":"Banach manifold","score":0.48453372716903687},{"id":"https://openalex.org/keywords/dual-space","display_name":"Dual space","score":0.4557868540287018},{"id":"https://openalex.org/keywords/closed-graph-theorem","display_name":"Closed graph theorem","score":0.4516989588737488},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.42277365922927856},{"id":"https://openalex.org/keywords/dual-pair","display_name":"Dual pair","score":0.41812416911125183},{"id":"https://openalex.org/keywords/functional-analysis","display_name":"Functional analysis","score":0.40959298610687256},{"id":"https://openalex.org/keywords/danskins-theorem","display_name":"Danskin's theorem","score":0.17260831594467163},{"id":"https://openalex.org/keywords/fixed-point-theorem","display_name":"Fixed-point theorem","score":0.1645270586013794},{"id":"https://openalex.org/keywords/brouwer-fixed-point-theorem","display_name":"Brouwer fixed-point theorem","score":0.16232669353485107}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8392942547798157},{"id":"https://openalex.org/C6183050","wikidata":"https://www.wikidata.org/wiki/Q3058180","display_name":"Interpolation space","level":4,"score":0.6889162063598633},{"id":"https://openalex.org/C33549677","wikidata":"https://www.wikidata.org/wiki/Q2032198","display_name":"Reflexive space","level":5,"score":0.6872791647911072},{"id":"https://openalex.org/C172294467","wikidata":"https://www.wikidata.org/wiki/Q305936","display_name":"Lp space","level":3,"score":0.6806066632270813},{"id":"https://openalex.org/C117145882","wikidata":"https://www.wikidata.org/wiki/Q1471397","display_name":"Fr\u00e9chet space","level":5,"score":0.6682255864143372},{"id":"https://openalex.org/C184908153","wikidata":"https://www.wikidata.org/wiki/Q2226650","display_name":"Eberlein\u2013\u0160mulian theorem","level":4,"score":0.6407216787338257},{"id":"https://openalex.org/C100022809","wikidata":"https://www.wikidata.org/wiki/Q5310221","display_name":"Dual norm","level":3,"score":0.6362659335136414},{"id":"https://openalex.org/C42030387","wikidata":"https://www.wikidata.org/wiki/Q944297","display_name":"Open mapping theorem (functional analysis)","level":5,"score":0.5871356129646301},{"id":"https://openalex.org/C132954091","wikidata":"https://www.wikidata.org/wiki/Q194397","display_name":"Banach space","level":2,"score":0.5859484672546387},{"id":"https://openalex.org/C169976997","wikidata":"https://www.wikidata.org/wiki/Q3058185","display_name":"Birnbaum\u2013Orlicz space","level":5,"score":0.5774690508842468},{"id":"https://openalex.org/C2780980858","wikidata":"https://www.wikidata.org/wiki/Q110022","display_name":"Dual (grammatical number)","level":2,"score":0.5741879343986511},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.5355809330940247},{"id":"https://openalex.org/C178099376","wikidata":"https://www.wikidata.org/wiki/Q7825044","display_name":"Topological tensor product","level":4,"score":0.4879915118217468},{"id":"https://openalex.org/C165593952","wikidata":"https://www.wikidata.org/wiki/Q3554819","display_name":"Banach manifold","level":4,"score":0.48453372716903687},{"id":"https://openalex.org/C45633534","wikidata":"https://www.wikidata.org/wiki/Q752487","display_name":"Dual space","level":2,"score":0.4557868540287018},{"id":"https://openalex.org/C20211961","wikidata":"https://www.wikidata.org/wiki/Q1068085","display_name":"Closed graph theorem","level":5,"score":0.4516989588737488},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.42277365922927856},{"id":"https://openalex.org/C198586994","wikidata":"https://www.wikidata.org/wiki/Q96376663","display_name":"Dual pair","level":5,"score":0.41812416911125183},{"id":"https://openalex.org/C55112680","wikidata":"https://www.wikidata.org/wiki/Q190549","display_name":"Functional analysis","level":3,"score":0.40959298610687256},{"id":"https://openalex.org/C48540410","wikidata":"https://www.wikidata.org/wiki/Q5221089","display_name":"Danskin's theorem","level":4,"score":0.17260831594467163},{"id":"https://openalex.org/C45962547","wikidata":"https://www.wikidata.org/wiki/Q1422068","display_name":"Fixed-point theorem","level":2,"score":0.1645270586013794},{"id":"https://openalex.org/C146147875","wikidata":"https://www.wikidata.org/wiki/Q1144897","display_name":"Brouwer fixed-point theorem","level":3,"score":0.16232669353485107},{"id":"https://openalex.org/C55493867","wikidata":"https://www.wikidata.org/wiki/Q7094","display_name":"Biochemistry","level":1,"score":0.0},{"id":"https://openalex.org/C142362112","wikidata":"https://www.wikidata.org/wiki/Q735","display_name":"Art","level":0,"score":0.0},{"id":"https://openalex.org/C104317684","wikidata":"https://www.wikidata.org/wiki/Q7187","display_name":"Gene","level":2,"score":0.0},{"id":"https://openalex.org/C185592680","wikidata":"https://www.wikidata.org/wiki/Q2329","display_name":"Chemistry","level":0,"score":0.0},{"id":"https://openalex.org/C124952713","wikidata":"https://www.wikidata.org/wiki/Q8242","display_name":"Literature","level":1,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.2478/forma-2014-0007","is_oa":true,"landing_page_url":"https://doi.org/10.2478/forma-2014-0007","pdf_url":"https://content.sciendo.com/downloadpdf/journals/forma/22/1/article-p69.pdf","source":{"id":"https://openalex.org/S179241345","display_name":"Formalized Mathematics","issn_l":"1426-2630","issn":["1426-2630","1898-9934"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320322","host_organization_name":"De Gruyter Open","host_organization_lineage":["https://openalex.org/P4310320322","https://openalex.org/P4310313990"],"host_organization_lineage_names":["De Gruyter Open","De Gruyter"],"type":"journal"},"license":"cc-by-sa","license_id":"https://openalex.org/licenses/cc-by-sa","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Formalized Mathematics","raw_type":"journal-article"},{"id":"pmh:oai:doaj.org/article:933244d0ef754b5bbd3095fcb2c581d5","is_oa":false,"landing_page_url":"https://doaj.org/article/933244d0ef754b5bbd3095fcb2c581d5","pdf_url":null,"source":{"id":"https://openalex.org/S4306401280","display_name":"DOAJ (DOAJ: Directory of Open Access Journals)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Formalized Mathematics, Vol 22, Iss 1, Pp 69-77 (2014)","raw_type":"article"},{"id":"pmh:oai:repozytorium.uwb.edu.pl:11320/3703","is_oa":false,"landing_page_url":"http://hdl.handle.net/11320/3703","pdf_url":null,"source":{"id":"https://openalex.org/S4306400933","display_name":"The Repository of University in Bialystok (University in Bialystok)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I4210094198","host_organization_name":"University Clinical Hospital In Bialystok","host_organization_lineage":["https://openalex.org/I4210094198"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Article"}],"best_oa_location":{"id":"doi:10.2478/forma-2014-0007","is_oa":true,"landing_page_url":"https://doi.org/10.2478/forma-2014-0007","pdf_url":"https://content.sciendo.com/downloadpdf/journals/forma/22/1/article-p69.pdf","source":{"id":"https://openalex.org/S179241345","display_name":"Formalized Mathematics","issn_l":"1426-2630","issn":["1426-2630","1898-9934"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320322","host_organization_name":"De Gruyter Open","host_organization_lineage":["https://openalex.org/P4310320322","https://openalex.org/P4310313990"],"host_organization_lineage_names":["De Gruyter Open","De Gruyter"],"type":"journal"},"license":"cc-by-sa","license_id":"https://openalex.org/licenses/cc-by-sa","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Formalized Mathematics","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W951606502.pdf","grobid_xml":"https://content.openalex.org/works/W951606502.grobid-xml"},"referenced_works_count":16,"referenced_works":["https://openalex.org/W66860599","https://openalex.org/W68895216","https://openalex.org/W173477520","https://openalex.org/W180239714","https://openalex.org/W192809544","https://openalex.org/W260340778","https://openalex.org/W2096867974","https://openalex.org/W2098464720","https://openalex.org/W2105559527","https://openalex.org/W2118265684","https://openalex.org/W2138253682","https://openalex.org/W2154026104","https://openalex.org/W2159772409","https://openalex.org/W2296232473","https://openalex.org/W2547540153","https://openalex.org/W4285719527"],"related_works":["https://openalex.org/W4248946191","https://openalex.org/W4244615838","https://openalex.org/W1553761754","https://openalex.org/W4241361679","https://openalex.org/W4234602485","https://openalex.org/W44522632","https://openalex.org/W4285222914","https://openalex.org/W40057417","https://openalex.org/W580130575","https://openalex.org/W951606502"],"abstract_inverted_index":{"Summary":[0],"In":[1,50],"this":[2],"article,":[3],"we":[4,16,30,52,66,81],"deal":[5],"with":[6],"dual":[7,18,32,44,61],"spaces":[8,19,23,33,45],"and":[9,24],"the":[10,14,40,56,64,71,78,83],"Hahn-Banach":[11,84],"Theorem.":[12],"At":[13],"first,":[15],"defined":[17,31],"of":[20,34,46,55,60,77],"real":[21,35,47,88],"linear":[22,48],"proved":[25,53,82],"related":[26],"basic":[27],"properties.":[28],"Next,":[29],"normed":[36,89],"spaces.":[37,49,62,90],"We":[38,91],"formed":[39],"definitions":[41],"based":[42],"on":[43],"addition,":[51],"properties":[54],"norm":[57],"about":[58],"elements":[59],"For":[63],"proof":[65],"referred":[67],"to":[68],"descriptions":[69],"in":[70,87],"article":[72],"[21].":[73],"Finally,":[74],"applying":[75],"theorems":[76],"second":[79],"section,":[80],"extension":[85],"theorem":[86],"have":[92],"used":[93,95],"extensively":[94],"[17].":[96]},"counts_by_year":[{"year":2024,"cited_by_count":1},{"year":2017,"cited_by_count":1},{"year":2015,"cited_by_count":3},{"year":2014,"cited_by_count":2}],"updated_date":"2026-06-11T09:08:48.828518","created_date":"2025-10-10T00:00:00"}
