{"id":"https://openalex.org/W2945358351","doi":"https://doi.org/10.2478/forma-2019-0002","title":"Bilinear Operators on Normed Linear Spaces","display_name":"Bilinear Operators on Normed Linear Spaces","publication_year":2019,"publication_date":"2019-04-01","ids":{"openalex":"https://openalex.org/W2945358351","doi":"https://doi.org/10.2478/forma-2019-0002","mag":"2945358351"},"language":"en","primary_location":{"id":"doi:10.2478/forma-2019-0002","is_oa":true,"landing_page_url":"https://doi.org/10.2478/forma-2019-0002","pdf_url":"https://content.sciendo.com/downloadpdf/journals/forma/27/1/article-p15.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/27/1/article-p15.pdf","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5005167371","display_name":"Kazuhisa Nakasho","orcid":"https://orcid.org/0000-0003-1110-4342"},"institutions":[{"id":"https://openalex.org/I173915773","display_name":"Yamaguchi University","ror":"https://ror.org/03cxys317","country_code":"JP","type":"education","lineage":["https://openalex.org/I173915773"]}],"countries":["JP"],"is_corresponding":true,"raw_author_name":"Kazuhisa Nakasho","raw_affiliation_strings":["Yamaguchi University , Yamaguchi , Japan"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Yamaguchi University , Yamaguchi , Japan","institution_ids":["https://openalex.org/I173915773"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5005167371"],"corresponding_institution_ids":["https://openalex.org/I173915773"],"apc_list":null,"apc_paid":null,"fwci":0.3247,"has_fulltext":true,"cited_by_count":2,"citation_normalized_percentile":{"value":0.49808719,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":94},"biblio":{"volume":"27","issue":"1","first_page":"15","last_page":"23"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12037","display_name":"Algebraic and Geometric Analysis","score":0.9535999894142151,"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/T12037","display_name":"Algebraic and Geometric Analysis","score":0.9535999894142151,"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/T10622","display_name":"Quantum Mechanics and Applications","score":0.9498000144958496,"subfield":{"id":"https://openalex.org/subfields/3107","display_name":"Atomic and Molecular Physics, and Optics"},"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/T10237","display_name":"Cryptography and Data Security","score":0.946399986743927,"subfield":{"id":"https://openalex.org/subfields/1702","display_name":"Artificial Intelligence"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.8267309069633484},{"id":"https://openalex.org/keywords/bilinear-interpolation","display_name":"Bilinear interpolation","score":0.6558902263641357},{"id":"https://openalex.org/keywords/operator-theory","display_name":"Operator theory","score":0.604127049446106},{"id":"https://openalex.org/keywords/banach-space","display_name":"Banach space","score":0.5813000798225403},{"id":"https://openalex.org/keywords/normed-vector-space","display_name":"Normed vector space","score":0.5717523694038391},{"id":"https://openalex.org/keywords/bilinear-map","display_name":"Bilinear map","score":0.5695568919181824},{"id":"https://openalex.org/keywords/algebraic-number","display_name":"Algebraic number","score":0.5353894829750061},{"id":"https://openalex.org/keywords/bounded-function","display_name":"Bounded function","score":0.5337349772453308},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.5272140502929688},{"id":"https://openalex.org/keywords/bounded-operator","display_name":"Bounded operator","score":0.519494891166687},{"id":"https://openalex.org/keywords/linear-operators","display_name":"Linear operators","score":0.46917039155960083},{"id":"https://openalex.org/keywords/continuous-linear-operator","display_name":"Continuous linear operator","score":0.46613141894340515},{"id":"https://openalex.org/keywords/algebra-over-a-field","display_name":"Algebra over a field","score":0.4468291699886322},{"id":"https://openalex.org/keywords/space","display_name":"Space (punctuation)","score":0.4449041485786438},{"id":"https://openalex.org/keywords/operator-norm","display_name":"Operator norm","score":0.43629738688468933},{"id":"https://openalex.org/keywords/lp-space","display_name":"Lp space","score":0.4272438585758209},{"id":"https://openalex.org/keywords/bilinear-form","display_name":"Bilinear form","score":0.4254280626773834},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.4028231203556061},{"id":"https://openalex.org/keywords/vector-space","display_name":"Vector space","score":0.31848663091659546},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.21017217636108398},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.10370400547981262}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8267309069633484},{"id":"https://openalex.org/C205203396","wikidata":"https://www.wikidata.org/wiki/Q612143","display_name":"Bilinear interpolation","level":2,"score":0.6558902263641357},{"id":"https://openalex.org/C43929395","wikidata":"https://www.wikidata.org/wiki/Q1198874","display_name":"Operator theory","level":2,"score":0.604127049446106},{"id":"https://openalex.org/C132954091","wikidata":"https://www.wikidata.org/wiki/Q194397","display_name":"Banach space","level":2,"score":0.5813000798225403},{"id":"https://openalex.org/C171739935","wikidata":"https://www.wikidata.org/wiki/Q726210","display_name":"Normed vector space","level":2,"score":0.5717523694038391},{"id":"https://openalex.org/C142685569","wikidata":"https://www.wikidata.org/wiki/Q1086961","display_name":"Bilinear map","level":3,"score":0.5695568919181824},{"id":"https://openalex.org/C9376300","wikidata":"https://www.wikidata.org/wiki/Q168817","display_name":"Algebraic number","level":2,"score":0.5353894829750061},{"id":"https://openalex.org/C34388435","wikidata":"https://www.wikidata.org/wiki/Q2267362","display_name":"Bounded function","level":2,"score":0.5337349772453308},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.5272140502929688},{"id":"https://openalex.org/C175454919","wikidata":"https://www.wikidata.org/wiki/Q2342396","display_name":"Bounded operator","level":3,"score":0.519494891166687},{"id":"https://openalex.org/C3019018024","wikidata":"https://www.wikidata.org/wiki/Q207643","display_name":"Linear operators","level":3,"score":0.46917039155960083},{"id":"https://openalex.org/C109175616","wikidata":"https://www.wikidata.org/wiki/Q1091024","display_name":"Continuous linear operator","level":2,"score":0.46613141894340515},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.4468291699886322},{"id":"https://openalex.org/C2778572836","wikidata":"https://www.wikidata.org/wiki/Q380933","display_name":"Space (punctuation)","level":2,"score":0.4449041485786438},{"id":"https://openalex.org/C68386474","wikidata":"https://www.wikidata.org/wiki/Q980557","display_name":"Operator norm","level":3,"score":0.43629738688468933},{"id":"https://openalex.org/C172294467","wikidata":"https://www.wikidata.org/wiki/Q305936","display_name":"Lp space","level":3,"score":0.4272438585758209},{"id":"https://openalex.org/C8828549","wikidata":"https://www.wikidata.org/wiki/Q837924","display_name":"Bilinear form","level":2,"score":0.4254280626773834},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.4028231203556061},{"id":"https://openalex.org/C13336665","wikidata":"https://www.wikidata.org/wiki/Q125977","display_name":"Vector space","level":2,"score":0.31848663091659546},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.21017217636108398},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.10370400547981262},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.0},{"id":"https://openalex.org/C111919701","wikidata":"https://www.wikidata.org/wiki/Q9135","display_name":"Operating system","level":1,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.2478/forma-2019-0002","is_oa":true,"landing_page_url":"https://doi.org/10.2478/forma-2019-0002","pdf_url":"https://content.sciendo.com/downloadpdf/journals/forma/27/1/article-p15.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:5abaeddfab4245ebb50af0366f15b8a6","is_oa":false,"landing_page_url":"https://doaj.org/article/5abaeddfab4245ebb50af0366f15b8a6","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 27, Iss 1, Pp 15-23 (2019)","raw_type":"article"},{"id":"pmh:oai:repozytorium.uwb.edu.pl:11320/7836","is_oa":true,"landing_page_url":"http://hdl.handle.net/11320/7836","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:10.2478/forma-2019-0002","is_oa":true,"landing_page_url":"https://doi.org/10.2478/forma-2019-0002","pdf_url":"https://content.sciendo.com/downloadpdf/journals/forma/27/1/article-p15.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":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2945358351.pdf","grobid_xml":"https://content.openalex.org/works/W2945358351.grobid-xml"},"referenced_works_count":4,"referenced_works":["https://openalex.org/W2044555352","https://openalex.org/W2526342011","https://openalex.org/W2770430340","https://openalex.org/W2915799038"],"related_works":["https://openalex.org/W3123914979","https://openalex.org/W2794086404","https://openalex.org/W2035999820","https://openalex.org/W2028111557","https://openalex.org/W2379426594","https://openalex.org/W2915799038","https://openalex.org/W4381805055","https://openalex.org/W2389343521","https://openalex.org/W2118722273","https://openalex.org/W4298313109"],"abstract_inverted_index":{"Summary":[0],"The":[1],"main":[2],"aim":[3],"of":[4,10,20,31,42,64,87,93,95],"this":[5,110],"article":[6],"is":[7,50,58,98],"proving":[8],"properties":[9],"bilinear":[11,32,44,66,88],"operators":[12,33,45,67,89],"on":[13,34,46],"normed":[14,47],"linear":[15,35,48,96],"spaces":[16,36,49],"formalized":[17],"by":[18],"means":[19],"Mizar":[21],"[1].":[22],"In":[23,53,78],"the":[24,40,54,61,79,82,85,91],"first":[25],"two":[26],"chapters,":[27],"algebraic":[28,62],"structures":[29],"[3]":[30],"are":[37],"discussed.":[38],"Especially,":[39],"space":[41,72,86,92],"bounded":[43,65],"developed":[51],"here.":[52],"third":[55],"chapter,":[56,81],"it":[57],"remarked":[59],"that":[60],"structure":[63],"to":[68,102],"a":[69,75],"certain":[70],"Banach":[71,76],"also":[73],"constitutes":[74],"space.":[77],"last":[80],"correspondence":[83],"between":[84],"and":[90,107],"composition":[94],"opearators":[97],"shown.":[99],"We":[100],"referred":[101],"[4],":[103],"[11],":[104],"[2],":[105],"[7]":[106],"[8]":[108],"in":[109],"formalization.":[111]},"counts_by_year":[{"year":2024,"cited_by_count":1},{"year":2019,"cited_by_count":1}],"updated_date":"2026-05-21T06:26:12.895304","created_date":"2025-10-10T00:00:00"}