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[{"key": "dc.contributor.advisor", "value": "Schultz, Timo", "language": null, "element": "contributor", "qualifier": "advisor", "schema": "dc"}, {"key": "dc.contributor.advisor", "value": "Eriksson-Bique, Sylvester", "language": null, "element": "contributor", "qualifier": "advisor", "schema": "dc"}, {"key": "dc.contributor.author", "value": "Halttunen, Miika", "language": null, "element": "contributor", "qualifier": "author", "schema": "dc"}, {"key": "dc.date.accessioned", "value": "2025-06-09T06:43:34Z", "language": null, "element": "date", "qualifier": "accessioned", "schema": "dc"}, {"key": "dc.date.available", "value": "2025-06-09T06:43:34Z", "language": null, "element": "date", "qualifier": "available", "schema": "dc"}, {"key": "dc.date.issued", "value": "2025", "language": null, "element": "date", "qualifier": "issued", "schema": "dc"}, {"key": "dc.identifier.uri", "value": "https://jyx.jyu.fi/handle/123456789/103216", "language": null, "element": "identifier", "qualifier": "uri", "schema": "dc"}, {"key": "dc.description.abstract", "value": "T\u00e4ss\u00e4 pro gradu -tutkielmassa k\u00e4sitell\u00e4\u00e4n pituusavaruuksia. Topologiseen\navaruuteen X voidaan ensin m\u00e4\u00e4ritell\u00e4 kuvaus L, jota kutsutaan pituusstruktuuriksi. Sen avulla voidaan m\u00e4\u00e4ritell\u00e4 pituusmetriikka d_L. Se m\u00e4\u00e4ritell\u00e4\u00e4n infimumina kaksi pistett\u00e4 x, y \u2208 X yhdist\u00e4vien polkujen pituuksista.\nMetrisell\u00e4 avaruudella tarkoitetaan puolestaan paria (X, d), jossa X on jokin joukko ja d joukkoon X m\u00e4\u00e4ritelty kahden pisteen v\u00e4linen et\u00e4isyys. Jos\non olemassa pituusstruktuuri L siten, ett\u00e4 d = d_L, niin metrist\u00e4 avaruutta\n(X, d) kutsutaan pituusavaruudeksi.\nPituusavaruudessa lyhimm\u00e4ll\u00e4 polulla tarkoitetaan sellaista polkua, jonka\npituus on sama kuin sen p\u00e4\u00e4tepisteiden v\u00e4linen et\u00e4isyys. Tutkielmassa tarkastellaan, mit\u00e4 ehtoja avaruuden X pit\u00e4\u00e4 toteuttaa, jotta kahden pisteen\nv\u00e4lill\u00e4 olisi olemassa lyhin polku. P\u00e4\u00e4tuloksena tutkielmassa muotoillaan ja\ntodistetaan Hopf\u2013Rinow\u2013Cohn-Vossen lause, joka kertoo, mitk\u00e4 ehdot pituusavaruuden X pit\u00e4\u00e4 toteuttaa, jotta kahden pisteen v\u00e4lill\u00e4 on olemassa lyhin polku. T\u00e4ll\u00f6in avaruutta X kutsutaan geodeettiseksi avaruudeksi.\nHopf\u2013Rinow\u2013Cohn-Vossen lauseen todistuksen j\u00e4lkeen annetaan kaksi esimerkki\u00e4, jotka osoittavat, ett\u00e4 kyseisess\u00e4 lauseessa esiintyv\u00e4t oletukset ovat\ntarpeellisia.\nTutkielman viimeisess\u00e4 luvussa osoitetaan tulos, ett\u00e4 jos (X, d) on t\u00e4ydellinen pituusavaruus, johon on m\u00e4\u00e4ritelty lokaalisti tuplaava mitta, niin (X, d)\non geodeettinen avaruus. Ensin tarkastellaan kuitenkin tuplaavia mittoja ja\nosoitetaan niiden avulla avaruuden X tuplaavuus. T\u00e4ll\u00f6in lokaalisti tuplaavan mitan tapaus on helpommin ymm\u00e4rrett\u00e4viss\u00e4.", "language": "fi", "element": "description", "qualifier": "abstract", "schema": "dc"}, {"key": "dc.description.abstract", "value": "", "language": "en", "element": "description", "qualifier": "abstract", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Submitted by Paivi Vuorio (paelvuor@jyu.fi) on 2025-06-09T06:43:34Z\nNo. of bitstreams: 0", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Made available in DSpace on 2025-06-09T06:43:34Z (GMT). No. of bitstreams: 0\n Previous issue date: 2025", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.format.extent", "value": "49", "language": null, "element": "format", "qualifier": "extent", "schema": "dc"}, {"key": "dc.language.iso", "value": "fin", "language": null, "element": "language", "qualifier": "iso", "schema": "dc"}, {"key": "dc.rights", "value": "In Copyright", "language": null, "element": "rights", "qualifier": null, "schema": "dc"}, {"key": "dc.subject.other", "value": "pituusavaruudet", "language": null, "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "pituusstruktuuri", "language": null, "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "lyhin polku", "language": null, "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.title", "value": "Hopf\u2013Rinow\u2013Cohn-Vossen lause", "language": null, "element": "title", "qualifier": null, "schema": "dc"}, {"key": "dc.type", "value": "master thesis", "language": null, "element": "type", "qualifier": null, "schema": "dc"}, {"key": "dc.identifier.urn", "value": "URN:NBN:fi:jyu-202506095020", "language": null, "element": "identifier", "qualifier": "urn", "schema": "dc"}, {"key": "dc.contributor.faculty", "value": "Matemaattis-luonnontieteellinen tiedekunta", "language": "fi", "element": "contributor", "qualifier": "faculty", "schema": "dc"}, {"key": "dc.contributor.faculty", "value": "Faculty of Sciences", "language": "en", "element": "contributor", "qualifier": "faculty", "schema": "dc"}, {"key": "dc.contributor.department", "value": "Matematiikan ja tilastotieteen laitos", "language": "fi", "element": "contributor", "qualifier": "department", "schema": "dc"}, {"key": "dc.contributor.department", "value": "Department of Mathematics and Statistics", "language": "en", "element": "contributor", "qualifier": "department", "schema": "dc"}, {"key": "dc.contributor.organization", "value": "Jyv\u00e4skyl\u00e4n yliopisto", "language": null, "element": "contributor", "qualifier": "organization", "schema": "dc"}, {"key": "dc.contributor.organization", "value": "University of Jyv\u00e4skyl\u00e4", "language": null, "element": "contributor", "qualifier": "organization", "schema": "dc"}, {"key": "dc.subject.discipline", "value": "Matematiikka", "language": "fi", "element": "subject", "qualifier": "discipline", "schema": "dc"}, {"key": "dc.subject.discipline", "value": "Mathematics", "language": "en", "element": "subject", "qualifier": "discipline", "schema": "dc"}, {"key": "dc.type.coar", "value": "http://purl.org/coar/resource_type/c_bdcc", "language": null, "element": "type", "qualifier": "coar", "schema": "dc"}, {"key": "dc.rights.copyright", "value": "\u00a9 The Author(s)", "language": "fi", "element": "rights", "qualifier": "copyright", "schema": "dc"}, {"key": "dc.rights.accesslevel", "value": "openAccess", "language": null, "element": "rights", "qualifier": "accesslevel", "schema": "dc"}, {"key": "dc.type.publication", "value": "masterThesis", "language": null, "element": "type", "qualifier": "publication", "schema": "dc"}, {"key": "dc.subject.yso", "value": "matematiikka", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "metriset avaruudet", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "topologiset avaruudet", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.rights.url", "value": "https://rightsstatements.org/page/InC/1.0/", "language": null, "element": "rights", "qualifier": "url", "schema": "dc"}, {"key": "dc.description.accessibilityfeature", "value": "unknown accessibility", "language": "en", "element": "description", "qualifier": "accessibilityfeature", "schema": "dc"}, {"key": "dc.description.accessibilityfeature", "value": "ei tietoa saavutettavuudesta", "language": "fi", "element": "description", "qualifier": "accessibilityfeature", "schema": "dc"}]
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