{"id":"https://openalex.org/W2294649372","doi":"https://doi.org/10.1515/forma-2015-0020","title":"The Orthogonal Projection and the Riesz Representation Theorem","display_name":"The Orthogonal Projection and the Riesz Representation Theorem","publication_year":2015,"publication_date":"2015-09-01","ids":{"openalex":"https://openalex.org/W2294649372","doi":"https://doi.org/10.1515/forma-2015-0020","mag":"2294649372"},"language":"en","primary_location":{"id":"doi:10.1515/forma-2015-0020","is_oa":true,"landing_page_url":"https://doi.org/10.1515/forma-2015-0020","pdf_url":"https://sciendo.com/pdf/10.1515/forma-2015-0020","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://sciendo.com/pdf/10.1515/forma-2015-0020","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/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"]}],"countries":["JP"],"is_corresponding":false,"raw_author_name":"Keiko Narita","raw_affiliation_strings":["Hirosaki-city Aomori, Japan"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Hirosaki-city Aomori, Japan","institution_ids":["https://openalex.org/I4210149837"]}]},{"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/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"]}],"countries":["JP"],"is_corresponding":false,"raw_author_name":"Noboru Endou","raw_affiliation_strings":["Gifu National College of Technology Gifu, Japan"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Gifu National College of Technology Gifu, Japan","institution_ids":["https://openalex.org/I4210113483"]}]},{"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":1,"citation_normalized_percentile":{"value":0.11289615,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":94},"biblio":{"volume":"23","issue":"3","first_page":"243","last_page":"252"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T13080","display_name":"Advanced Differential Geometry Research","score":0.9934999942779541,"subfield":{"id":"https://openalex.org/subfields/3103","display_name":"Astronomy and Astrophysics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T13080","display_name":"Advanced Differential Geometry Research","score":0.9934999942779541,"subfield":{"id":"https://openalex.org/subfields/3103","display_name":"Astronomy and Astrophysics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10545","display_name":"Optimization and Variational Analysis","score":0.9904000163078308,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11777","display_name":"Fixed Point Theorems Analysis","score":0.9890000224113464,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.8942795395851135},{"id":"https://openalex.org/keywords/hilbert-space","display_name":"Hilbert space","score":0.5863017439842224},{"id":"https://openalex.org/keywords/riesz-representation-theorem","display_name":"Riesz representation theorem","score":0.5834499597549438},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.5603694319725037},{"id":"https://openalex.org/keywords/m.-riesz-extension-theorem","display_name":"M. Riesz extension theorem","score":0.4778311252593994},{"id":"https://openalex.org/keywords/norm","display_name":"Norm (philosophy)","score":0.449584037065506},{"id":"https://openalex.org/keywords/topological-tensor-product","display_name":"Topological tensor product","score":0.44104260206222534},{"id":"https://openalex.org/keywords/hilbert-manifold","display_name":"Hilbert manifold","score":0.43984439969062805},{"id":"https://openalex.org/keywords/lp-space","display_name":"Lp space","score":0.41566166281700134},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.37115800380706787},{"id":"https://openalex.org/keywords/banach-space","display_name":"Banach space","score":0.25678467750549316},{"id":"https://openalex.org/keywords/functional-analysis","display_name":"Functional analysis","score":0.22282594442367554}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8942795395851135},{"id":"https://openalex.org/C62799726","wikidata":"https://www.wikidata.org/wiki/Q190056","display_name":"Hilbert space","level":2,"score":0.5863017439842224},{"id":"https://openalex.org/C113675457","wikidata":"https://www.wikidata.org/wiki/Q1357684","display_name":"Riesz representation theorem","level":2,"score":0.5834499597549438},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.5603694319725037},{"id":"https://openalex.org/C56969512","wikidata":"https://www.wikidata.org/wiki/Q3527217","display_name":"M. Riesz extension theorem","level":2,"score":0.4778311252593994},{"id":"https://openalex.org/C191795146","wikidata":"https://www.wikidata.org/wiki/Q3878446","display_name":"Norm (philosophy)","level":2,"score":0.449584037065506},{"id":"https://openalex.org/C178099376","wikidata":"https://www.wikidata.org/wiki/Q7825044","display_name":"Topological tensor product","level":4,"score":0.44104260206222534},{"id":"https://openalex.org/C147493315","wikidata":"https://www.wikidata.org/wiki/Q3554824","display_name":"Hilbert manifold","level":3,"score":0.43984439969062805},{"id":"https://openalex.org/C172294467","wikidata":"https://www.wikidata.org/wiki/Q305936","display_name":"Lp space","level":3,"score":0.41566166281700134},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.37115800380706787},{"id":"https://openalex.org/C132954091","wikidata":"https://www.wikidata.org/wiki/Q194397","display_name":"Banach space","level":2,"score":0.25678467750549316},{"id":"https://openalex.org/C55112680","wikidata":"https://www.wikidata.org/wiki/Q190549","display_name":"Functional analysis","level":3,"score":0.22282594442367554},{"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/C55493867","wikidata":"https://www.wikidata.org/wiki/Q7094","display_name":"Biochemistry","level":1,"score":0.0},{"id":"https://openalex.org/C17744445","wikidata":"https://www.wikidata.org/wiki/Q36442","display_name":"Political science","level":0,"score":0.0},{"id":"https://openalex.org/C199539241","wikidata":"https://www.wikidata.org/wiki/Q7748","display_name":"Law","level":1,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.1515/forma-2015-0020","is_oa":true,"landing_page_url":"https://doi.org/10.1515/forma-2015-0020","pdf_url":"https://sciendo.com/pdf/10.1515/forma-2015-0020","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:5d6464586c434abfa0e28e4ad0f928ad","is_oa":false,"landing_page_url":"https://doaj.org/article/5d6464586c434abfa0e28e4ad0f928ad","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 23, Iss 3, Pp 243-252 (2015)","raw_type":"article"},{"id":"pmh:oai:repozytorium.uwb.edu.pl:11320/4898","is_oa":false,"landing_page_url":"http://hdl.handle.net/11320/4898","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:10.1515/forma-2015-0020","is_oa":true,"landing_page_url":"https://doi.org/10.1515/forma-2015-0020","pdf_url":"https://sciendo.com/pdf/10.1515/forma-2015-0020","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":[{"id":"https://openalex.org/G5886570192","display_name":"Content Development for Distance Education in Advanced University Mathematics Using Mizar","funder_award_id":"22300285","funder_id":"https://openalex.org/F4320334764","funder_display_name":"Japan Society for the Promotion of Science"},{"id":"https://openalex.org/G7250689446","display_name":"Construction of reasoning library for realization of computer verification system","funder_award_id":"23500029","funder_id":"https://openalex.org/F4320334764","funder_display_name":"Japan Society for the Promotion of Science"}],"funders":[{"id":"https://openalex.org/F4320334764","display_name":"Japan Society for the Promotion of Science","ror":"https://ror.org/00hhkn466"}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2294649372.pdf","grobid_xml":"https://content.openalex.org/works/W2294649372.grobid-xml"},"referenced_works_count":22,"referenced_works":["https://openalex.org/W66860599","https://openalex.org/W173477520","https://openalex.org/W180239714","https://openalex.org/W192809544","https://openalex.org/W260340778","https://openalex.org/W951606502","https://openalex.org/W1545761024","https://openalex.org/W2096867974","https://openalex.org/W2098464720","https://openalex.org/W2102061347","https://openalex.org/W2105559527","https://openalex.org/W2118265684","https://openalex.org/W2129703334","https://openalex.org/W2154026104","https://openalex.org/W2159772409","https://openalex.org/W2163641194","https://openalex.org/W2296232473","https://openalex.org/W2341439142","https://openalex.org/W2577060450","https://openalex.org/W3038830718","https://openalex.org/W3186605155","https://openalex.org/W4285719527"],"related_works":["https://openalex.org/W2035270346","https://openalex.org/W2595960242","https://openalex.org/W4244773120","https://openalex.org/W4236314116","https://openalex.org/W3166000218","https://openalex.org/W1838837447","https://openalex.org/W4230469657","https://openalex.org/W4287766370","https://openalex.org/W4365185544","https://openalex.org/W1526769101"],"abstract_inverted_index":{"Abstract":[0],"In":[1,15,127,165],"this":[2,42],"article,":[3],"the":[4,8,16,21,68,71,73,83,109,131,138,146,157,166,173,183,189,196,199,218],"orthogonal":[5,139,147],"projection":[6],"and":[7,29,57,70,95,163,168,195,215],"Riesz":[9,190],"representation":[10,191],"theorem":[11],"are":[12,118,156],"mainly":[13],"formalized.":[14],"first":[17],"section,":[18,170],"we":[19,44,66,86,114,134,142,171,187],"defined":[20,30,67,88,172],"norm":[22,200],"of":[23,47,53,61,75,103,111,121,137,149,151,161,175,198,201],"elements":[24,150],"on":[25,77,91,123,145,178,205],"real":[26,34,38,48,54,62,78,92,104,124,152,179,206],"Hilbert":[27,39,49,63,79,93,105,153,180,207],"spaces,":[28,56],"Mizar":[31],"functor":[32],"RUSp2RNSp,":[33],"normed":[35,55,125],"spaces":[36,50,94,102],"as":[37,51],"spaces.":[40,64,80,106,126,154,181,208],"By":[41,182],"definition,":[43],"regarded":[45],"sequences":[46,52],"proved":[58,96,115,143],"some":[59,89,97],"properties":[60,110,120,136],"Furthermore,":[65],"continuity":[69,74],"Lipschitz":[72],"functionals":[76,122,177,204],"Referring":[81],"to":[82,108,211],"article":[84],"[15],":[85],"also":[87],"definitions":[90,132],"theorems":[98,144,160],"for":[99],"defining":[100],"dual":[101],"As":[107],"all":[112],"definitions,":[113],"that":[116],"they":[117],"equivalent":[119],"Sec.":[128],"2,":[129],"by":[130],"[11],":[133],"showed":[135,188],"complement.":[140],"Then":[141],"decomposition":[148],"They":[155],"last":[158,184],"two":[159],"existence":[162],"uniqueness.":[164],"third":[167],"final":[169],"kernel":[174],"linear":[176,203],"three":[185],"theorems,":[186],"theorem,":[192],"existence,":[193],"uniqueness,":[194],"property":[197],"bounded":[202],"We":[209],"referred":[210],"[36],":[212],"[9],":[213],"[24]":[214],"[3]":[216],"in":[217],"formalization.":[219]},"counts_by_year":[{"year":2024,"cited_by_count":1}],"updated_date":"2026-06-11T09:08:48.828518","created_date":"2025-10-10T00:00:00"}