{"id":"https://openalex.org/W7207773522","doi":"https://doi.org/10.48550/arxiv.2609.01885","title":"The Order of the Non-universal Tree of a Hilbert Space with Respect to the Haar Basis","display_name":"The Order of the Non-universal Tree of a Hilbert Space with Respect to the Haar Basis","publication_year":2026,"publication_date":"2026-09-01","ids":{"openalex":"https://openalex.org/W7207773522","doi":"https://doi.org/10.48550/arxiv.2609.01885"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2609.01885","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.01885","pdf_url":"https://arxiv.org/pdf/2609.01885","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"volume":null,"type":"preprint","is_traditional":true,"indexed_in":["arxiv","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/2609.01885","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5149939057","display_name":"Sam Whitmire","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I4210119109","display_name":"Indiana University Bloomington","ror":"https://ror.org/02k40bc56","country_code":"US","type":"education","lineage":["https://openalex.org/I4210119109","https://openalex.org/I592451"]},{"id":"https://openalex.org/I592451","display_name":"Indiana University","ror":"https://ror.org/01kg8sb98","country_code":"US","type":"education","lineage":["https://openalex.org/I592451"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"Whitmire, Sam","raw_affiliation_strings":["Indiana University Bloomington Department of Mathematics , Bloomington , IN 47405"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Indiana University Bloomington Department of Mathematics , Bloomington , IN 47405","institution_ids":["https://openalex.org/I4210119109","https://openalex.org/I592451"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5149939057"],"corresponding_institution_ids":["https://openalex.org/I4210119109","https://openalex.org/I592451"],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":null,"cited_by_percentile_year":null,"biblio":{"volume":null,"issue":null,"first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11545","display_name":"Advanced Banach Space Theory","score":0.9990000128746033,"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":0.9990000128746033,"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/T11151","display_name":"Advanced Topology and Set Theory","score":9.999999747378752e-05,"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/T10899","display_name":"Advanced Operator Algebra Research","score":9.999999747378752e-05,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/separable-banach-spaces","display_name":"separable Banach spaces","score":0.847000002861023},{"id":"https://openalex.org/keywords/hilbert-spaces","display_name":"Hilbert spaces","score":0.8029999732971191},{"id":"https://openalex.org/keywords/haar-basis","display_name":"Haar basis","score":0.9850000143051147}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8278999924659729},{"id":"https://openalex.org/C70710897","wikidata":"https://www.wikidata.org/wiki/Q680081","display_name":"Separable space","level":2,"score":0.7057999968528748},{"id":"https://openalex.org/C132954091","wikidata":"https://www.wikidata.org/wiki/Q194397","display_name":"Banach space","level":2,"score":0.5501000285148621},{"id":"https://openalex.org/C12426560","wikidata":"https://www.wikidata.org/wiki/Q189569","display_name":"Basis (linear algebra)","level":2,"score":0.5458999872207642},{"id":"https://openalex.org/C12362212","wikidata":"https://www.wikidata.org/wiki/Q728435","display_name":"Linear subspace","level":2,"score":0.5389000177383423},{"id":"https://openalex.org/C62799726","wikidata":"https://www.wikidata.org/wiki/Q190056","display_name":"Hilbert space","level":2,"score":0.5382000207901001},{"id":"https://openalex.org/C76103427","wikidata":"https://www.wikidata.org/wiki/Q5761197","display_name":"Hilbert R-tree","level":5,"score":0.4925000071525574},{"id":"https://openalex.org/C203436722","wikidata":"https://www.wikidata.org/wiki/Q902950","display_name":"Isomorphism (crystallography)","level":3,"score":0.4668000042438507},{"id":"https://openalex.org/C2777212361","wikidata":"https://www.wikidata.org/wiki/Q5127848","display_name":"Class (philosophy)","level":2,"score":0.4629000127315521},{"id":"https://openalex.org/C113174947","wikidata":"https://www.wikidata.org/wiki/Q2859736","display_name":"Tree (set theory)","level":2,"score":0.450300008058548},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.44609999656677246},{"id":"https://openalex.org/C147493315","wikidata":"https://www.wikidata.org/wiki/Q3554824","display_name":"Hilbert manifold","level":3,"score":0.4318999946117401},{"id":"https://openalex.org/C182306322","wikidata":"https://www.wikidata.org/wiki/Q1779371","display_name":"Order (exchange)","level":2,"score":0.40869998931884766},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.3896999955177307},{"id":"https://openalex.org/C80739010","wikidata":"https://www.wikidata.org/wiki/Q6029720","display_name":"Infinite-dimensional vector function","level":5,"score":0.37790000438690186},{"id":"https://openalex.org/C185982889","wikidata":"https://www.wikidata.org/wiki/Q5761255","display_name":"Hilbert\u2013Poincar\u00e9 series","level":2,"score":0.36959999799728394},{"id":"https://openalex.org/C2778572836","wikidata":"https://www.wikidata.org/wiki/Q380933","display_name":"Space (punctuation)","level":2,"score":0.353300005197525},{"id":"https://openalex.org/C36185835","wikidata":"https://www.wikidata.org/wiki/Q3058230","display_name":"Projective Hilbert space","level":4,"score":0.35100001096725464},{"id":"https://openalex.org/C60064506","wikidata":"https://www.wikidata.org/wiki/Q1262627","display_name":"Rigged Hilbert space","level":4,"score":0.31630000472068787},{"id":"https://openalex.org/C76801838","wikidata":"https://www.wikidata.org/wiki/Q777879","display_name":"Schauder basis","level":3,"score":0.3133000135421753},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.29739999771118164},{"id":"https://openalex.org/C172294467","wikidata":"https://www.wikidata.org/wiki/Q305936","display_name":"Lp space","level":3,"score":0.28299999237060547},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.2761000096797943},{"id":"https://openalex.org/C80884492","wikidata":"https://www.wikidata.org/wiki/Q3345678","display_name":"Reproducing kernel Hilbert space","level":3,"score":0.2563000023365021},{"id":"https://openalex.org/C187029792","wikidata":"https://www.wikidata.org/wiki/Q2179112","display_name":"Haar","level":3,"score":0.25529998540878296},{"id":"https://openalex.org/C57493831","wikidata":"https://www.wikidata.org/wiki/Q3134666","display_name":"Projection (relational algebra)","level":2,"score":0.2551000118255615},{"id":"https://openalex.org/C40753290","wikidata":"https://www.wikidata.org/wiki/Q1139111","display_name":"Quotient space (topology)","level":3,"score":0.2524000108242035}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2609.01885","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.01885","pdf_url":"https://arxiv.org/pdf/2609.01885","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"doi:10.48550/arxiv.2609.01885","doi":"https://doi.org/10.48550/arxiv.2609.01885","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2609.01885","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2609.01885","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.01885","pdf_url":"https://arxiv.org/pdf/2609.01885","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"sustainable_development_goals":[],"study_designs":[],"awards":[],"funders":[],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7207773522.pdf","grobid_xml":"https://content.openalex.org/works/W7207773522.grobid-xml"},"referenced_works_count":0,"referenced_works":[],"related_works":["https://openalex.org/W2289310713","https://openalex.org/W2790486302","https://openalex.org/W2010759686","https://openalex.org/W7219155405","https://openalex.org/W2216888106","https://openalex.org/W7098881155","https://openalex.org/W2060278062","https://openalex.org/W338558486","https://openalex.org/W2963926617","https://openalex.org/W1984026590"],"abstract_inverted_index":{"In":[0],"his":[1],"1994":[2],"doctoral":[3],"thesis,":[4],"Bossard":[5],"introduced":[6],"the":[7,10,32,45,52,81,84,114,121,127],"notion":[8],"of":[9,28,43,47,54,56,59,64,73,77,80,83,123],"non-universal":[11,85],"tree":[12,86],"$T_{NU}(X)$":[13],"associated":[14],"with":[15,31,111,131],"each":[16],"separable":[17,48,100],"Banach":[18,49],"space":[19,90],"$X$":[20],"which":[21],"does":[22],"not":[23],"contain":[24],"an":[25],"isomorphic":[26],"copy":[27],"$C(2^\\mathbb{N})$.":[29,60],"Together":[30],"order":[33,82],"operation":[34],"defined":[35],"on":[36],"well-founded":[37],"trees,":[38],"we":[39,70],"obtain":[40],"a":[41,88],"method":[42],"classifying":[44],"complexity":[46],"spaces":[50,125],"by":[51],"degree":[53],"isomorphism":[55],"finite-dimensional":[57],"subspaces":[58],"Despite":[61],"further":[62],"refinement":[63],"this":[65,134,138],"concept":[66],"in":[67],"later":[68],"years,":[69],"are":[71],"unaware":[72],"any":[74,99],"specific":[75],"instances":[76],"direct":[78],"exhibitions":[79],"for":[87,117],"concrete":[89],"and":[91],"basis.":[92,139],"We":[93],"show":[94],"that":[95,120],"if":[96],"$H$":[97],"is":[98,126],"Hilbert":[101,124],"space,":[102],"then":[103],"$o(T_{NU}(H))":[104],"=":[105],"\u03c9":[106],"+":[107],"1$":[108],"when":[109,136],"taken":[110],"respect":[112,132],"to":[113,133],"Haar":[115],"basis":[116],"$C(2^\\mathbb{N})$,":[118],"demonstrating":[119],"class":[122,130],"least":[128],"complex":[129],"measurement":[135],"considering":[137]},"counts_by_year":[],"updated_date":"2026-11-06T12:05:46.655746","created_date":"2026-09-04T00:00:00"}
