{"id":"https://openalex.org/W1145362253","doi":"https://doi.org/10.2478/forma-2014-0024","title":"Topological Properties of Real Normed Space","display_name":"Topological Properties of Real Normed Space","publication_year":2014,"publication_date":"2014-03-31","ids":{"openalex":"https://openalex.org/W1145362253","doi":"https://doi.org/10.2478/forma-2014-0024","mag":"1145362253"},"language":"en","primary_location":{"id":"doi:10.2478/forma-2014-0024","is_oa":true,"landing_page_url":"https://doi.org/10.2478/forma-2014-0024","pdf_url":null,"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://doi.org/10.2478/forma-2014-0024","any_repository_has_fulltext":false},"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/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":"Kazuhisa Nakasho","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":"middle","author":{"id":"https://openalex.org/A5076839736","display_name":"Yuichi Futa","orcid":null},"institutions":[{"id":"https://openalex.org/I177738480","display_name":"Japan Advanced Institute of Science and Technology","ror":"https://ror.org/03frj4r98","country_code":"JP","type":"education","lineage":["https://openalex.org/I177738480"]}],"countries":["JP"],"is_corresponding":false,"raw_author_name":"Yuichi Futa","raw_affiliation_strings":["Japan Advanced Institute of Science and Technology Ishikawa, Japan","Japan Advanced Institute of Science and Technology, Ishikawa, Japan"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Japan Advanced Institute of Science and Technology Ishikawa, Japan","institution_ids":["https://openalex.org/I177738480"]},{"raw_affiliation_string":"Japan Advanced Institute of Science and Technology, Ishikawa, Japan","institution_ids":["https://openalex.org/I177738480"]}]},{"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":1.0081,"has_fulltext":false,"cited_by_count":4,"citation_normalized_percentile":{"value":0.76484913,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":96},"biblio":{"volume":"22","issue":"3","first_page":"209","last_page":"223"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12062","display_name":"Approximation Theory and Sequence Spaces","score":0.9902999997138977,"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"}},"topics":[{"id":"https://openalex.org/T12062","display_name":"Approximation Theory and Sequence Spaces","score":0.9902999997138977,"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"}},{"id":"https://openalex.org/T11777","display_name":"Fixed Point Theorems Analysis","score":0.9714000225067139,"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"}},{"id":"https://openalex.org/T10545","display_name":"Optimization and Variational Analysis","score":0.965399980545044,"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/normed-vector-space","display_name":"Normed vector space","score":0.8023393750190735},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.7488270401954651},{"id":"https://openalex.org/keywords/linear-subspace","display_name":"Linear subspace","score":0.6739255785942078},{"id":"https://openalex.org/keywords/banach-space","display_name":"Banach space","score":0.5501372814178467},{"id":"https://openalex.org/keywords/quotient-space","display_name":"Quotient space (topology)","score":0.4987819194793701},{"id":"https://openalex.org/keywords/lipschitz-continuity","display_name":"Lipschitz continuity","score":0.48109668493270874},{"id":"https://openalex.org/keywords/topological-vector-space","display_name":"Topological vector space","score":0.48090696334838867},{"id":"https://openalex.org/keywords/kernel","display_name":"Kernel (algebra)","score":0.4657163619995117},{"id":"https://openalex.org/keywords/reflexive-space","display_name":"Reflexive space","score":0.4318937361240387},{"id":"https://openalex.org/keywords/convergence","display_name":"Convergence (economics)","score":0.4224732220172882},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.3938748836517334},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.383976012468338},{"id":"https://openalex.org/keywords/topological-space","display_name":"Topological space","score":0.3575267791748047},{"id":"https://openalex.org/keywords/quotient","display_name":"Quotient","score":0.2774507999420166},{"id":"https://openalex.org/keywords/interpolation-space","display_name":"Interpolation space","score":0.24226495623588562},{"id":"https://openalex.org/keywords/functional-analysis","display_name":"Functional analysis","score":0.22205880284309387}],"concepts":[{"id":"https://openalex.org/C171739935","wikidata":"https://www.wikidata.org/wiki/Q726210","display_name":"Normed vector space","level":2,"score":0.8023393750190735},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7488270401954651},{"id":"https://openalex.org/C12362212","wikidata":"https://www.wikidata.org/wiki/Q728435","display_name":"Linear subspace","level":2,"score":0.6739255785942078},{"id":"https://openalex.org/C132954091","wikidata":"https://www.wikidata.org/wiki/Q194397","display_name":"Banach space","level":2,"score":0.5501372814178467},{"id":"https://openalex.org/C40753290","wikidata":"https://www.wikidata.org/wiki/Q1139111","display_name":"Quotient space (topology)","level":3,"score":0.4987819194793701},{"id":"https://openalex.org/C22324862","wikidata":"https://www.wikidata.org/wiki/Q652707","display_name":"Lipschitz continuity","level":2,"score":0.48109668493270874},{"id":"https://openalex.org/C47776270","wikidata":"https://www.wikidata.org/wiki/Q1455249","display_name":"Topological vector space","level":3,"score":0.48090696334838867},{"id":"https://openalex.org/C74193536","wikidata":"https://www.wikidata.org/wiki/Q574844","display_name":"Kernel (algebra)","level":2,"score":0.4657163619995117},{"id":"https://openalex.org/C33549677","wikidata":"https://www.wikidata.org/wiki/Q2032198","display_name":"Reflexive space","level":5,"score":0.4318937361240387},{"id":"https://openalex.org/C2777303404","wikidata":"https://www.wikidata.org/wiki/Q759757","display_name":"Convergence (economics)","level":2,"score":0.4224732220172882},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.3938748836517334},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.383976012468338},{"id":"https://openalex.org/C81332173","wikidata":"https://www.wikidata.org/wiki/Q179899","display_name":"Topological space","level":2,"score":0.3575267791748047},{"id":"https://openalex.org/C199422724","wikidata":"https://www.wikidata.org/wiki/Q41118","display_name":"Quotient","level":2,"score":0.2774507999420166},{"id":"https://openalex.org/C6183050","wikidata":"https://www.wikidata.org/wiki/Q3058180","display_name":"Interpolation space","level":4,"score":0.24226495623588562},{"id":"https://openalex.org/C55112680","wikidata":"https://www.wikidata.org/wiki/Q190549","display_name":"Functional analysis","level":3,"score":0.22205880284309387},{"id":"https://openalex.org/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display_name":"Economics","level":0,"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/C50522688","wikidata":"https://www.wikidata.org/wiki/Q189833","display_name":"Economic growth","level":1,"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/C104317684","wikidata":"https://www.wikidata.org/wiki/Q7187","display_name":"Gene","level":2,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.2478/forma-2014-0024","is_oa":true,"landing_page_url":"https://doi.org/10.2478/forma-2014-0024","pdf_url":null,"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:d8c70e130f9f492592f8a9ed514d2d07","is_oa":false,"landing_page_url":"https://doaj.org/article/d8c70e130f9f492592f8a9ed514d2d07","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 22, Iss 3, Pp 209-223 (2014)","raw_type":"article"},{"id":"pmh:oai:repozytorium.uwb.edu.pl:11320/3720","is_oa":false,"landing_page_url":"http://hdl.handle.net/11320/3720","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-2014-0024","is_oa":true,"landing_page_url":"https://doi.org/10.2478/forma-2014-0024","pdf_url":null,"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":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":20,"referenced_works":["https://openalex.org/W22141270","https://openalex.org/W192809544","https://openalex.org/W951606502","https://openalex.org/W2096867974","https://openalex.org/W2098464720","https://openalex.org/W2099252149","https://openalex.org/W2112849387","https://openalex.org/W2118265684","https://openalex.org/W2126555322","https://openalex.org/W2133311781","https://openalex.org/W2143779468","https://openalex.org/W2154026104","https://openalex.org/W2159772409","https://openalex.org/W2163641194","https://openalex.org/W2296232473","https://openalex.org/W2304533832","https://openalex.org/W2341439142","https://openalex.org/W2740943156","https://openalex.org/W3038830718","https://openalex.org/W4285719527"],"related_works":["https://openalex.org/W2486282296","https://openalex.org/W4251690266","https://openalex.org/W4250479592","https://openalex.org/W2075707935","https://openalex.org/W103014951","https://openalex.org/W4236145557","https://openalex.org/W1982767646","https://openalex.org/W260796275","https://openalex.org/W4206375061","https://openalex.org/W3198469467"],"abstract_inverted_index":{"Summary":[0],"In":[1,12,95],"this":[2],"article,":[3],"we":[4,29,37,78,99],"formalize":[5,91],"topological":[6],"properties":[7,31,102],"of":[8,32,47,71,103,106],"real":[9,33,42,83,107],"normed":[10,34,43,84,108],"spaces.":[11,88],"the":[13,80,96,101,104],"first":[14],"part,":[15],"open":[16],"and":[17,21,23,57,117],"closed,":[18],"density,":[19],"separability":[20],"sequence":[22],"its":[24],"convergence":[25,70],"are":[26,60,112],"discussed.":[27],"Then":[28,36,77],"argue":[30,79,100],"subspace.":[35],"discuss":[38],"linear":[39,51],"functions":[40,52],"between":[41],"speces.":[44],"Several":[45],"kinds":[46],"subspaces":[48,85],"induced":[49],"by":[50],"such":[53],"as":[54],"kernel,":[55],"image":[56,59],"inverse":[58],"considered":[61],"here.":[62,76],"The":[63],"fact":[64],"that":[65],"Lipschitz":[66],"continuity":[67],"operators":[68],"preserve":[69],"sequences":[72],"is":[73],"also":[74,90],"refered":[75],"condition":[81],"when":[82],"become":[86],"Banach\u2019s":[87],"We":[89],"quotient":[92],"vector":[93],"space.":[94,109],"last":[97],"session,":[98],"closure":[105],"These":[110],"formalizations":[111],"based":[113],"on":[114],"[19](p.3-41),":[115],"[2]":[116],"[34](p.3-67).":[118]},"counts_by_year":[{"year":2018,"cited_by_count":1},{"year":2015,"cited_by_count":2},{"year":2014,"cited_by_count":1}],"updated_date":"2026-06-11T09:08:48.828518","created_date":"2025-10-10T00:00:00"}
