{"id":"https://openalex.org/W2779493305","doi":"https://doi.org/10.1515/forma-2017-0017","title":"F. Riesz Theorem","display_name":"F. Riesz Theorem","publication_year":2017,"publication_date":"2017-10-01","ids":{"openalex":"https://openalex.org/W2779493305","doi":"https://doi.org/10.1515/forma-2017-0017","mag":"2779493305"},"language":"en","primary_location":{"id":"doi:10.1515/forma-2017-0017","is_oa":true,"landing_page_url":"https://doi.org/10.1515/forma-2017-0017","pdf_url":"https://content.sciendo.com/downloadpdf/journals/forma/25/3/article-p179.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/25/3/article-p179.pdf","any_repository_has_fulltext":true},"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/I5467274","display_name":"Akita Prefectural University","ror":"https://ror.org/05b1kx621","country_code":"JP","type":"education","lineage":["https://openalex.org/I5467274"]}],"countries":["JP"],"is_corresponding":false,"raw_author_name":"Keiko Narita","raw_affiliation_strings":["Hirosaki-city, Aomori , Japan","Shinshu University Nagano, Japan","Akita Prefectural University Akita, Japan"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Hirosaki-city, Aomori , Japan","institution_ids":[]},{"raw_affiliation_string":"Shinshu University Nagano, Japan","institution_ids":["https://openalex.org/I137975476"]},{"raw_affiliation_string":"Akita Prefectural University Akita, Japan","institution_ids":["https://openalex.org/I5467274"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5005167371","display_name":"Kazuhisa Nakasho","orcid":"https://orcid.org/0000-0003-1110-4342"},"institutions":[{"id":"https://openalex.org/I5467274","display_name":"Akita Prefectural University","ror":"https://ror.org/05b1kx621","country_code":"JP","type":"education","lineage":["https://openalex.org/I5467274"]}],"countries":["JP"],"is_corresponding":false,"raw_author_name":"Kazuhisa Nakasho","raw_affiliation_strings":["Akita Prefectural University , Akita , Japan"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Akita Prefectural University , Akita , Japan","institution_ids":["https://openalex.org/I5467274"]}]},{"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":0.0,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":{"value":0.23070627,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"25","issue":"3","first_page":"179","last_page":"184"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12508","display_name":"Functional Equations Stability Results","score":0.9983999729156494,"subfield":{"id":"https://openalex.org/subfields/2604","display_name":"Applied Mathematics"},"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/T12508","display_name":"Functional Equations Stability Results","score":0.9983999729156494,"subfield":{"id":"https://openalex.org/subfields/2604","display_name":"Applied Mathematics"},"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/T11564","display_name":"Mathematical Inequalities and Applications","score":0.9965000152587891,"subfield":{"id":"https://openalex.org/subfields/2604","display_name":"Applied Mathematics"},"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/T12062","display_name":"Approximation Theory and Sequence Spaces","score":0.9957000017166138,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.8751001358032227},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.5364492535591125},{"id":"https://openalex.org/keywords/interval","display_name":"Interval (graph theory)","score":0.5113932490348816},{"id":"https://openalex.org/keywords/closed-graph-theorem","display_name":"Closed graph theorem","score":0.4933694899082184},{"id":"https://openalex.org/keywords/section","display_name":"Section (typography)","score":0.48752471804618835},{"id":"https://openalex.org/keywords/m.-riesz-extension-theorem","display_name":"M. Riesz extension theorem","score":0.48277345299720764},{"id":"https://openalex.org/keywords/riesz-representation-theorem","display_name":"Riesz representation theorem","score":0.45083606243133545},{"id":"https://openalex.org/keywords/banach-space","display_name":"Banach space","score":0.4495759904384613},{"id":"https://openalex.org/keywords/unit-interval","display_name":"Unit interval","score":0.41924387216567993},{"id":"https://openalex.org/keywords/bounded-function","display_name":"Bounded function","score":0.4179668724536896},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.413047730922699},{"id":"https://openalex.org/keywords/real-number","display_name":"Real number","score":0.4101465940475464},{"id":"https://openalex.org/keywords/fixed-point-theorem","display_name":"Fixed-point theorem","score":0.3506065607070923},{"id":"https://openalex.org/keywords/brouwer-fixed-point-theorem","display_name":"Brouwer fixed-point theorem","score":0.3263552784919739},{"id":"https://openalex.org/keywords/danskins-theorem","display_name":"Danskin's theorem","score":0.22373726963996887},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.20210248231887817},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.14750730991363525}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8751001358032227},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.5364492535591125},{"id":"https://openalex.org/C2778067643","wikidata":"https://www.wikidata.org/wiki/Q166507","display_name":"Interval (graph theory)","level":2,"score":0.5113932490348816},{"id":"https://openalex.org/C20211961","wikidata":"https://www.wikidata.org/wiki/Q1068085","display_name":"Closed graph theorem","level":5,"score":0.4933694899082184},{"id":"https://openalex.org/C2780129039","wikidata":"https://www.wikidata.org/wiki/Q1931107","display_name":"Section (typography)","level":2,"score":0.48752471804618835},{"id":"https://openalex.org/C56969512","wikidata":"https://www.wikidata.org/wiki/Q3527217","display_name":"M. Riesz extension theorem","level":2,"score":0.48277345299720764},{"id":"https://openalex.org/C113675457","wikidata":"https://www.wikidata.org/wiki/Q1357684","display_name":"Riesz representation theorem","level":2,"score":0.45083606243133545},{"id":"https://openalex.org/C132954091","wikidata":"https://www.wikidata.org/wiki/Q194397","display_name":"Banach space","level":2,"score":0.4495759904384613},{"id":"https://openalex.org/C128394911","wikidata":"https://www.wikidata.org/wiki/Q1987578","display_name":"Unit interval","level":2,"score":0.41924387216567993},{"id":"https://openalex.org/C34388435","wikidata":"https://www.wikidata.org/wiki/Q2267362","display_name":"Bounded function","level":2,"score":0.4179668724536896},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.413047730922699},{"id":"https://openalex.org/C94020503","wikidata":"https://www.wikidata.org/wiki/Q12916","display_name":"Real number","level":2,"score":0.4101465940475464},{"id":"https://openalex.org/C45962547","wikidata":"https://www.wikidata.org/wiki/Q1422068","display_name":"Fixed-point theorem","level":2,"score":0.3506065607070923},{"id":"https://openalex.org/C146147875","wikidata":"https://www.wikidata.org/wiki/Q1144897","display_name":"Brouwer fixed-point theorem","level":3,"score":0.3263552784919739},{"id":"https://openalex.org/C48540410","wikidata":"https://www.wikidata.org/wiki/Q5221089","display_name":"Danskin's theorem","level":4,"score":0.22373726963996887},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.20210248231887817},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.14750730991363525},{"id":"https://openalex.org/C144133560","wikidata":"https://www.wikidata.org/wiki/Q4830453","display_name":"Business","level":0,"score":0.0},{"id":"https://openalex.org/C112698675","wikidata":"https://www.wikidata.org/wiki/Q37038","display_name":"Advertising","level":1,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.1515/forma-2017-0017","is_oa":true,"landing_page_url":"https://doi.org/10.1515/forma-2017-0017","pdf_url":"https://content.sciendo.com/downloadpdf/journals/forma/25/3/article-p179.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:3e729e107c9847cf848a94fb6ea4ac4f","is_oa":false,"landing_page_url":"https://doaj.org/article/3e729e107c9847cf848a94fb6ea4ac4f","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 25, Iss 3, Pp 179-184 (2017)","raw_type":"article"},{"id":"pmh:oai:repozytorium.uwb.edu.pl:11320/6292","is_oa":true,"landing_page_url":"http://hdl.handle.net/11320/6292","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":"cc-by-sa","license_id":"https://openalex.org/licenses/cc-by-sa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Article"}],"best_oa_location":{"id":"doi:10.1515/forma-2017-0017","is_oa":true,"landing_page_url":"https://doi.org/10.1515/forma-2017-0017","pdf_url":"https://content.sciendo.com/downloadpdf/journals/forma/25/3/article-p179.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/W2779493305.pdf","grobid_xml":"https://content.openalex.org/works/W2779493305.grobid-xml"},"referenced_works_count":15,"referenced_works":["https://openalex.org/W608087936","https://openalex.org/W935710377","https://openalex.org/W951606502","https://openalex.org/W1545761024","https://openalex.org/W1855754411","https://openalex.org/W2042076004","https://openalex.org/W2143224343","https://openalex.org/W2184838578","https://openalex.org/W2554707563","https://openalex.org/W2577060450","https://openalex.org/W2588347594","https://openalex.org/W2592770093","https://openalex.org/W2753614965","https://openalex.org/W2758703457","https://openalex.org/W2912565176"],"related_works":["https://openalex.org/W2035270346","https://openalex.org/W2892015605","https://openalex.org/W2440474188","https://openalex.org/W4294755053","https://openalex.org/W4294226274","https://openalex.org/W2443005407","https://openalex.org/W2597522874","https://openalex.org/W3142231895","https://openalex.org/W2554233620","https://openalex.org/W1155039162"],"abstract_inverted_index":{"Summary":[0],"In":[1,16,87,100],"this":[2,145],"article,":[3],"we":[4,20,51,89,102,147],"formalize":[5],"in":[6,130],"the":[7,12,17,38,44,48,53,68,94,108,112,133,136,140,151,155],"Mizar":[8,22],"system":[9],"[1,":[10],"4]":[11],"F.":[13,97,104],"Riesz":[14,98,105],"theorem.":[15,99,138],"first":[18],"section,":[19,146],"defined":[21,52,67],"functor":[23],"ClstoCmp":[24],",":[25],"compact":[26],"topological":[27],"spaces":[28,55,70],"as":[29],"closed":[30,60,75,118],"interval":[31,61,76,119],"subset":[32,62,77,120],"of":[33,56,63,71,78,96,111,114,121,135,142,144],"real":[34,64,79,122],"numbers.":[35,80],"Then":[36],"using":[37],"former":[39],"definition":[40],"and":[41,47,66,154,166],"referring":[42],"to":[43,132,150],"article":[45,49,152,156],"[10]":[46],"[5],":[50],"normed":[54,69],"continuous":[57,115],"functions":[58,73,116],"on":[59,74,117,162],"numbers,":[65,123],"bounded":[72],"We":[81,125],"also":[82,148],"proved":[83,90,103],"some":[84,91],"related":[85],"properties.":[86],"Sec.2,":[88],"lemmas":[92],"for":[93],"proof":[95,134],"Sec.3,":[101],"theorem,":[106],"about":[107],"dual":[109],"space":[110,113],"finally.":[124],"applied":[126],"Hahn-Banach":[127],"theorem":[128],"(36)":[129],"[7],":[131],"last":[137],"For":[139],"description":[141],"theorems":[143],"referred":[149],"[8]":[153],"[6].":[157],"These":[158],"formalizations":[159],"are":[160],"based":[161],"[2],":[163],"[3],":[164],"[9],":[165],"[11].":[167]},"counts_by_year":[],"updated_date":"2026-06-11T09:08:48.828518","created_date":"2018-01-05T00:00:00"}