{"id":"https://openalex.org/W4387907401","doi":"https://doi.org/10.3390/math11214405","title":"Inequalities That Imply the Norm of a Linear Space Is Induced by an Inner Product","display_name":"Inequalities That Imply the Norm of a Linear Space Is Induced by an Inner Product","publication_year":2023,"publication_date":"2023-10-24","ids":{"openalex":"https://openalex.org/W4387907401","doi":"https://doi.org/10.3390/math11214405"},"language":"en","primary_location":{"id":"doi:10.3390/math11214405","doi":"https://doi.org/10.3390/math11214405","is_oa":true,"landing_page_url":"https://doi.org/10.3390/math11214405","pdf_url":"https://www.mdpi.com/2227-7390/11/21/4405/pdf?version=1698138219","source":{"id":"https://openalex.org/S4210192031","display_name":"Mathematics","issn_l":"2227-7390","issn":["2227-7390"],"is_oa":true,"is_in_doaj":true,"is_core":true,"listed_in":["anvur-scientific","cwts-core","doaj","poland-20","russia-white-list-1","vabb-shw"],"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Mathematics","raw_type":"journal-article"},"parent_works":[],"lineage":["https://openalex.org/W4387907401"],"child_works_count":0,"type":"article","is_traditional":true,"indexed_in":["crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://www.mdpi.com/2227-7390/11/21/4405/pdf?version=1698138219","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5089328870","display_name":"Sorin R\u0103dulescu","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I97553796","display_name":"University of Craiova","ror":"https://ror.org/03s251g81","country_code":"RO","type":"education","lineage":["https://openalex.org/I97553796"]}],"countries":["RO"],"is_corresponding":false,"raw_author_name":"Sorin R\u0103dulescu","raw_affiliation_strings":["Department of Mathematics, University of Craiova, 200585 Craiova, Romania"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, University of Craiova, 200585 Craiova, Romania","institution_ids":["https://openalex.org/I97553796"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5102774321","display_name":"Marius R\u0103dulescu","orcid":"https://orcid.org/0000-0003-4410-6133","observed_orcids":["https://orcid.org/0000-0003-4410-6133"]},"institutions":[{"id":"https://openalex.org/I58077936","display_name":"Romanian Academy","ror":"https://ror.org/0561n6946","country_code":"RO","type":"archive","lineage":["https://openalex.org/I58077936"]}],"countries":["RO"],"is_corresponding":true,"raw_author_name":"Marius R\u0103dulescu","raw_affiliation_strings":["\u201cGheorghe Mihoc\u2014Caius Iacob\u201d Institute of Mathematical Statistics and Applied Mathematics, Romanian Academy, Calea 13 Septembrie nr. 13, 050711 Bucharest, Romania"],"raw_orcid":"https://orcid.org/0000-0003-4410-6133","affiliations":[{"raw_affiliation_string":"\u201cGheorghe Mihoc\u2014Caius Iacob\u201d Institute of Mathematical Statistics and Applied Mathematics, Romanian Academy, Calea 13 Septembrie nr. 13, 050711 Bucharest, Romania","institution_ids":["https://openalex.org/I58077936"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5058101291","display_name":"Mih\u00e1ly Bencze","orcid":null,"observed_orcids":[]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Mihaly Bencze","raw_affiliation_strings":["Aprily Lajos High School, 500026 Brasov, Romania"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Aprily Lajos High School, 500026 Brasov, Romania","institution_ids":[]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5102774321"],"corresponding_institution_ids":["https://openalex.org/I58077936"],"apc_list":{"value":2100,"currency":"CHF","value_usd":2336},"apc_paid":{"value":2100,"currency":"CHF","value_usd":2336},"fwci":0.2349,"has_fulltext":true,"cited_by_count":1,"citation_normalized_percentile":{"value":0.33265843,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":93},"biblio":{"volume":"11","issue":"21","first_page":"4405","last_page":"4405"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11545","display_name":"Advanced Banach Space Theory","score":0.6269999742507935,"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"}},"topics":[{"id":"https://openalex.org/T11545","display_name":"Advanced Banach Space Theory","score":0.6269999742507935,"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"}},{"id":"https://openalex.org/T11564","display_name":"Mathematical Inequalities and Applications","score":0.30219998955726624,"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/T12504","display_name":"Mathematics and Applications","score":0.040800001472234726,"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/inner-product-spaces","display_name":"inner product spaces","score":0.8820000290870667},{"id":"https://openalex.org/keywords/normed-spaces","display_name":"normed spaces","score":0.8550000190734863},{"id":"https://openalex.org/keywords/characterization-of-inner-product-spaces","display_name":"characterization of inner product spaces","score":0.8870000243186951},{"id":"https://openalex.org/keywords/inequality-mathematics","display_name":"inequality (mathematics)","score":0.800000011920929},{"id":"https://openalex.org/keywords/tetrahedron","display_name":"tetrahedron","score":0.7770000100135803},{"id":"https://openalex.org/keywords/triangle-inequality","display_name":"triangle inequality","score":0.414000004529953}],"concepts":[{"id":"https://openalex.org/C91855343","wikidata":"https://www.wikidata.org/wiki/Q214159","display_name":"Inner product space","level":2,"score":0.9500457644462585},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8076008558273315},{"id":"https://openalex.org/C2776809875","wikidata":"https://www.wikidata.org/wiki/Q1375963","display_name":"Converse","level":2,"score":0.7761960029602051},{"id":"https://openalex.org/C191795146","wikidata":"https://www.wikidata.org/wiki/Q3878446","display_name":"Norm (philosophy)","level":2,"score":0.6708024740219116},{"id":"https://openalex.org/C105239961","wikidata":"https://www.wikidata.org/wiki/Q160003","display_name":"Tetrahedron","level":2,"score":0.5882076025009155},{"id":"https://openalex.org/C90673727","wikidata":"https://www.wikidata.org/wiki/Q901718","display_name":"Product (mathematics)","level":2,"score":0.5773012638092041},{"id":"https://openalex.org/C108710211","wikidata":"https://www.wikidata.org/wiki/Q11538","display_name":"Mathematical proof","level":2,"score":0.5638625621795654},{"id":"https://openalex.org/C2778572836","wikidata":"https://www.wikidata.org/wiki/Q380933","display_name":"Space (punctuation)","level":2,"score":0.5514182448387146},{"id":"https://openalex.org/C182964748","wikidata":"https://www.wikidata.org/wiki/Q208216","display_name":"Triangle inequality","level":2,"score":0.49805641174316406},{"id":"https://openalex.org/C171739935","wikidata":"https://www.wikidata.org/wiki/Q726210","display_name":"Normed vector space","level":2,"score":0.4879309833049774},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.46188846230506897},{"id":"https://openalex.org/C123222754","wikidata":"https://www.wikidata.org/wiki/Q1070472","display_name":"Product topology","level":2,"score":0.4551104009151459},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.4295716881752014},{"id":"https://openalex.org/C176370821","wikidata":"https://www.wikidata.org/wiki/Q1826459","display_name":"Linear space","level":2,"score":0.42322593927383423},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.30725449323654175},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.11827969551086426},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.09560033679008484},{"id":"https://openalex.org/C111919701","wikidata":"https://www.wikidata.org/wiki/Q9135","display_name":"Operating system","level":1,"score":0.0},{"id":"https://openalex.org/C199539241","wikidata":"https://www.wikidata.org/wiki/Q7748","display_name":"Law","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}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.3390/math11214405","doi":"https://doi.org/10.3390/math11214405","is_oa":true,"landing_page_url":"https://doi.org/10.3390/math11214405","pdf_url":"https://www.mdpi.com/2227-7390/11/21/4405/pdf?version=1698138219","source":{"id":"https://openalex.org/S4210192031","display_name":"Mathematics","issn_l":"2227-7390","issn":["2227-7390"],"is_oa":true,"is_in_doaj":true,"is_core":true,"listed_in":["anvur-scientific","cwts-core","doaj","poland-20","russia-white-list-1","vabb-shw"],"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Mathematics","raw_type":"journal-article"},{"id":"pmh:oai:RePEc:gam:jmathe:v:11:y:2023:i:21:p:4405-:d:1266016","doi":null,"is_oa":false,"landing_page_url":"https://www.mdpi.com/2227-7390/11/21/4405/","pdf_url":null,"source":{"id":"https://openalex.org/S4306401271","display_name":"RePEc: Research Papers in Economics","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I77793887","host_organization_name":"Federal Reserve Bank of St. Louis","host_organization_lineage":["https://openalex.org/I77793887"],"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"},{"id":"pmh:oai:doaj.org/article:f6c62823f1374b2db920d3a7bd260785","doi":null,"is_oa":true,"landing_page_url":"https://doaj.org/article/f6c62823f1374b2db920d3a7bd260785","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,"listed_in":[],"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"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":"Mathematics, Vol 11, Iss 21, p 4405 (2023)","raw_type":"article"}],"best_oa_location":{"id":"doi:10.3390/math11214405","doi":"https://doi.org/10.3390/math11214405","is_oa":true,"landing_page_url":"https://doi.org/10.3390/math11214405","pdf_url":"https://www.mdpi.com/2227-7390/11/21/4405/pdf?version=1698138219","source":{"id":"https://openalex.org/S4210192031","display_name":"Mathematics","issn_l":"2227-7390","issn":["2227-7390"],"is_oa":true,"is_in_doaj":true,"is_core":true,"listed_in":["anvur-scientific","cwts-core","doaj","poland-20","russia-white-list-1","vabb-shw"],"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Mathematics","raw_type":"journal-article"},"sustainable_development_goals":[],"study_designs":[],"awards":[],"funders":[],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W4387907401.pdf","grobid_xml":"https://content.openalex.org/works/W4387907401.grobid-xml"},"referenced_works_count":21,"referenced_works":["https://openalex.org/W113482007","https://openalex.org/W143363309","https://openalex.org/W319004803","https://openalex.org/W1518096223","https://openalex.org/W2028760356","https://openalex.org/W2082858262","https://openalex.org/W2088836904","https://openalex.org/W2092591518","https://openalex.org/W2321866085","https://openalex.org/W2322126400","https://openalex.org/W2324151815","https://openalex.org/W2327468149","https://openalex.org/W2331807671","https://openalex.org/W2334761136","https://openalex.org/W2765633503","https://openalex.org/W2782732335","https://openalex.org/W3035410284","https://openalex.org/W3093870245","https://openalex.org/W3184923569","https://openalex.org/W3200044139","https://openalex.org/W4256136326"],"related_works":["https://openalex.org/W2028760356","https://openalex.org/W2124547086","https://openalex.org/W1972426960","https://openalex.org/W2038929006","https://openalex.org/W2357373720","https://openalex.org/W2043096342","https://openalex.org/W3190171977","https://openalex.org/W2041714394","https://openalex.org/W2022485955","https://openalex.org/W2989347340"],"abstract_inverted_index":{"The":[0,91],"aim":[1],"of":[2,30,33,63,77,80,86,93,117,119,125,152],"this":[3,94,100,132],"paper":[4],"is":[5,13,56,67,109],"to":[6,144],"investigate":[7],"when":[8],"a":[9,21,81,113,126],"linear":[10,22],"normed":[11,23],"space":[12,24],"an":[14,59,71,87],"inner":[15,60,72,103,135],"product":[16,73,104,136],"space.":[17],"Several":[18],"conditions":[19,47],"in":[20,70,112],"are":[25,35,41,48,83,122],"formulated":[26],"with":[27,50],"the":[28,37,51,54,64,68,75,84,110,115,123,150],"help":[29],"inequalities.":[31],"Some":[32],"them":[34],"from":[36,149],"literature":[38],"and":[39],"others":[40],"new.":[42],"We":[43,128],"prove":[44,129],"that":[45,53,131],"these":[46],"equivalent":[49],"fact":[52],"norm":[55],"induced":[57],"by":[58],"product.":[61],"One":[62],"new":[65,107],"results":[66],"following:":[69,111],"space,":[74],"sum":[76,116],"opposite":[78,120],"edges":[79,121],"tetrahedron":[82],"sides":[85],"acute":[88],"angled":[89],"triangle.":[90,127],"converse":[92],"result":[95,108],"holds":[96],"also.":[97],"More":[98],"precisely,":[99],"property":[101,133],"characterizes":[102,134],"spaces.":[105,137],"Another":[106],"tetrahedron,":[114],"squares":[118],"lengths":[124],"also":[130],"In":[138],"addition,":[139],"we":[140],"give":[141],"simpler":[142],"proofs":[143],"some":[145],"theorems":[146],"already":[147],"known":[148],"publications":[151],"other":[153],"authors.":[154]},"counts_by_year":[{"year":2025,"cited_by_count":1}],"updated_date":"2026-10-16T09:59:22.235173","created_date":"2023-10-25T00:00:00"}
