{"id":"https://openalex.org/W4285326553","doi":"https://doi.org/10.2478/forma-2021-0022","title":"The 3-Fold Product Space of Real Normed Spaces and its Properties","display_name":"The 3-Fold Product Space of Real Normed Spaces and its Properties","publication_year":2021,"publication_date":"2021-12-01","ids":{"openalex":"https://openalex.org/W4285326553","doi":"https://doi.org/10.2478/forma-2021-0022"},"language":"en","primary_location":{"id":"doi:10.2478/forma-2021-0022","is_oa":true,"landing_page_url":"https://doi.org/10.2478/forma-2021-0022","pdf_url":"https://www.sciendo.com/pdf/10.2478/forma-2021-0022","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"],"open_access":{"is_oa":true,"oa_status":"diamond","oa_url":"https://www.sciendo.com/pdf/10.2478/forma-2021-0022","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5048322538","display_name":"Hiroyuki Okazaki","orcid":"https://orcid.org/0009-0000-1022-361X"},"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":"Hiroyuki Okazaki","raw_affiliation_strings":["Shinshu University , Nagano , Japan"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Shinshu University , Nagano , Japan","institution_ids":["https://openalex.org/I137975476"]}]},{"author_position":"last","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":false,"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":2,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":0.1448,"has_fulltext":true,"cited_by_count":1,"citation_normalized_percentile":{"value":0.55724455,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":94},"biblio":{"volume":"29","issue":"4","first_page":"241","last_page":"248"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11727","display_name":"Advanced Algebra and Logic","score":0.930899977684021,"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"}},"topics":[{"id":"https://openalex.org/T11727","display_name":"Advanced Algebra and Logic","score":0.930899977684021,"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/T11125","display_name":"Petri Nets in System Modeling","score":0.9151999950408936,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/cartesian-product","display_name":"Cartesian product","score":0.7940008640289307},{"id":"https://openalex.org/keywords/normed-vector-space","display_name":"Normed vector space","score":0.7305698394775391},{"id":"https://openalex.org/keywords/inner-product-space","display_name":"Inner product space","score":0.6818370223045349},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6756483912467957},{"id":"https://openalex.org/keywords/product","display_name":"Product (mathematics)","score":0.5761793851852417},{"id":"https://openalex.org/keywords/fold","display_name":"Fold (higher-order function)","score":0.5572413802146912},{"id":"https://openalex.org/keywords/function-space","display_name":"Function space","score":0.4976711571216583},{"id":"https://openalex.org/keywords/space","display_name":"Space (punctuation)","score":0.4963589310646057},{"id":"https://openalex.org/keywords/product-topology","display_name":"Product topology","score":0.46423736214637756},{"id":"https://openalex.org/keywords/algebra-over-a-field","display_name":"Algebra over a field","score":0.40157896280288696},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.3743056654930115},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.35187917947769165},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.24511945247650146},{"id":"https://openalex.org/keywords/programming-language","display_name":"Programming language","score":0.07497036457061768},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.06391480565071106}],"concepts":[{"id":"https://openalex.org/C65236422","wikidata":"https://www.wikidata.org/wiki/Q173740","display_name":"Cartesian product","level":2,"score":0.7940008640289307},{"id":"https://openalex.org/C171739935","wikidata":"https://www.wikidata.org/wiki/Q726210","display_name":"Normed vector space","level":2,"score":0.7305698394775391},{"id":"https://openalex.org/C91855343","wikidata":"https://www.wikidata.org/wiki/Q214159","display_name":"Inner product space","level":2,"score":0.6818370223045349},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6756483912467957},{"id":"https://openalex.org/C90673727","wikidata":"https://www.wikidata.org/wiki/Q901718","display_name":"Product (mathematics)","level":2,"score":0.5761793851852417},{"id":"https://openalex.org/C53942344","wikidata":"https://www.wikidata.org/wiki/Q951651","display_name":"Fold (higher-order function)","level":2,"score":0.5572413802146912},{"id":"https://openalex.org/C142730499","wikidata":"https://www.wikidata.org/wiki/Q934367","display_name":"Function space","level":2,"score":0.4976711571216583},{"id":"https://openalex.org/C2778572836","wikidata":"https://www.wikidata.org/wiki/Q380933","display_name":"Space (punctuation)","level":2,"score":0.4963589310646057},{"id":"https://openalex.org/C123222754","wikidata":"https://www.wikidata.org/wiki/Q1070472","display_name":"Product topology","level":2,"score":0.46423736214637756},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.40157896280288696},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.3743056654930115},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.35187917947769165},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.24511945247650146},{"id":"https://openalex.org/C199360897","wikidata":"https://www.wikidata.org/wiki/Q9143","display_name":"Programming language","level":1,"score":0.07497036457061768},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.06391480565071106},{"id":"https://openalex.org/C111919701","wikidata":"https://www.wikidata.org/wiki/Q9135","display_name":"Operating system","level":1,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.2478/forma-2021-0022","is_oa":true,"landing_page_url":"https://doi.org/10.2478/forma-2021-0022","pdf_url":"https://www.sciendo.com/pdf/10.2478/forma-2021-0022","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:repozytorium.uwb.edu.pl:11320/13668","is_oa":false,"landing_page_url":"http://hdl.handle.net/11320/13668","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-2021-0022","is_oa":true,"landing_page_url":"https://doi.org/10.2478/forma-2021-0022","pdf_url":"https://www.sciendo.com/pdf/10.2478/forma-2021-0022","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/W4285326553.pdf","grobid_xml":"https://content.openalex.org/works/W4285326553.grobid-xml"},"referenced_works_count":4,"referenced_works":["https://openalex.org/W2044555352","https://openalex.org/W2101610511","https://openalex.org/W2295728669","https://openalex.org/W2770430340"],"related_works":["https://openalex.org/W2953883784","https://openalex.org/W2015131486","https://openalex.org/W2587777217","https://openalex.org/W1968825347","https://openalex.org/W4384558110","https://openalex.org/W3092588299","https://openalex.org/W2181558739","https://openalex.org/W3176435573","https://openalex.org/W2392520244","https://openalex.org/W4249340230"],"abstract_inverted_index":{"Summary":[0],"In":[1],"this":[2],"article,":[3],"we":[4,48,63,76],"formalize":[5,64],"in":[6,20,41,94],"Mizar":[7,43],"[1],":[8],"[2]":[9],"the":[10,27,30,42,53,65,78,95],"3-fold":[11,57,68,81],"product":[12,32,59,69,82],"space":[13,33,70,83],"of":[14,29,34,67,71,80,84],"real":[15,35,72,85],"normed":[16,36,86],"spaces":[17,37],"for":[18,60],"usefulness":[19],"application":[21],"fields":[22],"such":[23],"as":[24],"engineering,":[25],"although":[26],"formalization":[28],"2-fold":[31],"has":[38],"been":[39],"stored":[40],"Mathematical":[44],"Library":[45],"[3].":[46],"First,":[47],"prove":[49],"some":[50],"theorems":[51],"about":[52],"3-variable":[54],"function":[55],"and":[56,92],"Cartesian":[58],"preparation.":[61],"Then":[62],"definition":[66,79],"linear":[73],"spaces.":[74,87],"Finally,":[75],"formulate":[77],"We":[88],"referred":[89],"to":[90],"[7]":[91],"[6]":[93],"formalization.":[96]},"counts_by_year":[{"year":2024,"cited_by_count":1}],"updated_date":"2026-05-21T06:26:12.895304","created_date":"2025-10-10T00:00:00"}