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[{"key": "dc.contributor.advisor", "value": "Julin, Vesa", "language": "", "element": "contributor", "qualifier": "advisor", "schema": "dc"}, {"key": "dc.contributor.author", "value": "Pajala, Topi", "language": "", "element": "contributor", "qualifier": "author", "schema": "dc"}, {"key": "dc.date.accessioned", "value": "2020-11-12T07:31:28Z", "language": null, "element": "date", "qualifier": "accessioned", "schema": "dc"}, {"key": "dc.date.available", "value": "2020-11-12T07:31:28Z", "language": null, "element": "date", "qualifier": "available", "schema": "dc"}, {"key": "dc.date.issued", "value": "2020", "language": "", "element": "date", "qualifier": "issued", "schema": "dc"}, {"key": "dc.identifier.uri", "value": "https://jyx.jyu.fi/handle/123456789/72574", "language": null, "element": "identifier", "qualifier": "uri", "schema": "dc"}, {"key": "dc.description.abstract", "value": "T\u00e4ss\u00e4 ty\u00f6ss\u00e4 tutkitaan Hilbertin avaruuksia, kompakteja operaattoreita Hilbertin avaruuksissa ja sit\u00e4, miten kompaktien operaattoreiden avulla on mahdollista muodostaa kanta Hilbertin avaruudelle. Kompakteilla operaattoreilla tarkoitetaan rajoitettuja lineaarikuvauksia, jotka kuvaavat jokaisen rajoitetun jonon sellaiseksi, ett\u00e4 sen kuvajoukosta l\u00f6ytyy osajono, joka suppenee. Tavallisesti \u00e4\u00e4rellisulotteiselle sis\u00e4tuloavaruudelle saadaan muodostettua kanta Hermiten operaattoreiden avulla, mutta \u00e4\u00e4ret\u00f6nulotteisen Hilbertin avaruuden tapauksessa l\u00e4hes t\u00e4ysin vastaava teoria l\u00f6ytyy kompakteista operaattoreista. P\u00e4\u00e4asiassa Hilbertin avaruuden kannan l\u00f6yt\u00e4miseksi riitt\u00e4\u00e4 l\u00f6yt\u00e4\u00e4 kompakti operaattori avaruudesta, jolloin kannan muodostavat ne avaruuden alkiot, jotka operaattori kuvaa samaksi alkioksi jollain reaaliluvulla kerrottuna.\n\nTutkielma koostuu nelj\u00e4st\u00e4 osasta, joista ensimm\u00e4isess\u00e4 tutustutaan Hilbertin avaruuteen ja sen rakenteeseen, toisessa osassa tutkitaan kompakteja operaattoreita yleisess\u00e4 Hilbertin avaruudessa ja osoitetaan, ett\u00e4 yleiselle Hilbertin avaruudelle on mahdollista muodostaa kanta kompaktien operaattoreiden avulla. Kolmannessa osassa m\u00e4\u00e4ritell\u00e4\u00e4n Sobolev-avaruudet ja tarkastellaan niiden yhteytt\u00e4 Hilbertin avaruuksiin ja nelj\u00e4nness\u00e4 osassa tutkitaan divergenssimuotoisia yht\u00e4l\u00f6it\u00e4 erityisesti sellaisissa avaruuksissa jotka ovat sek\u00e4 Hilbertin avaruuksia, ett\u00e4 Sobolev-avaruuksia.\n\nTutkielman p\u00e4\u00e4tuloksena osoitetaan, ett\u00e4 tiettyjen divergenssimuotoisten yht\u00e4l\u00f6iden ratkaisut ovat kompakteja operaattoreita ja edelleen n\u00e4iden avulla on mahdollista muodostaa koko avaruudelle kanta. Lopuksi osoitetaan, ett\u00e4 t\u00e4ll\u00e4 edell\u00e4 mainitulla menetelm\u00e4ll\u00e4 on mahdollista ratkaista helposti niin sanottu l\u00e4mp\u00f6yht\u00e4l\u00f6, joka kuvaa keskim\u00e4\u00e4r\u00e4ist\u00e4 l\u00e4mm\u00f6n jakautumista kappaleessa tietyll\u00e4 ajanhetkell\u00e4.", "language": "fi", "element": "description", "qualifier": "abstract", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Submitted by Paivi Vuorio (paelvuor@jyu.fi) on 2020-11-12T07:31:28Z\nNo. of bitstreams: 0", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Made available in DSpace on 2020-11-12T07:31:28Z (GMT). No. of bitstreams: 0\n Previous issue date: 2020", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.format.extent", "value": "61", "language": "", "element": "format", "qualifier": "extent", "schema": "dc"}, {"key": "dc.format.mimetype", "value": "application/pdf", "language": null, "element": "format", "qualifier": "mimetype", "schema": "dc"}, {"key": "dc.language.iso", "value": "fin", "language": null, "element": "language", "qualifier": "iso", "schema": "dc"}, {"key": "dc.rights", "value": "In Copyright", "language": "en", "element": "rights", "qualifier": null, "schema": "dc"}, {"key": "dc.subject.other", "value": "lineaarikuvaukset", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "sis\u00e4tuloavaruudet", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "kompaktit operaattorit", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.title", "value": "Hilbertin avaruudet ja kompaktit operaattorit", "language": "", "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-202011126608", "language": "", "element": "identifier", "qualifier": "urn", "schema": "dc"}, {"key": "dc.type.ontasot", "value": "Pro gradu -tutkielma", "language": "fi", "element": "type", "qualifier": "ontasot", "schema": "dc"}, {"key": "dc.type.ontasot", "value": "Master\u2019s thesis", "language": "en", "element": "type", "qualifier": "ontasot", "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": "fi", "element": "contributor", "qualifier": "organization", "schema": "dc"}, {"key": "dc.contributor.organization", "value": "University of Jyv\u00e4skyl\u00e4", "language": "en", "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": 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