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[{"key": "dc.contributor.author", "value": "Korhonen, Anni", "language": null, "element": "contributor", "qualifier": "author", "schema": "dc"}, {"key": "dc.date.accessioned", "value": "2025-02-19T09:45:04Z", "language": null, "element": "date", "qualifier": "accessioned", "schema": "dc"}, {"key": "dc.date.available", "value": "2025-02-19T09:45:04Z", "language": null, "element": "date", "qualifier": "available", "schema": "dc"}, {"key": "dc.date.issued", "value": "2023", "language": null, "element": "date", "qualifier": "issued", "schema": "dc"}, {"key": "dc.identifier.uri", "value": "https://jyx.jyu.fi/handle/123456789/100277", "language": null, "element": "identifier", "qualifier": "uri", "schema": "dc"}, {"key": "dc.description.abstract", "value": "T\u00e4ss\u00e4 pro gradu-tutkielmassa esitell\u00e4\u00e4n ja todistetaan Rieszin esityslause\nsek\u00e4 tarkastellaan sen soveltamista kalvon mekaniikan ongelmissa. Yksinkertaistetusti Rieszin esityslauseen mukaan Hilbertin avaruudessa, joka on t\u00e4ydellinen sis\u00e4tuloavaruus, jatkuvat lineaariset funktionaalit voidaan esitt\u00e4\u00e4 siten, ett\u00e4 funktionaalia vastaa yksik\u00e4sitteisen alkio, ja funktionaalin arvo on muuttujan sis\u00e4tulo kyseisen alkion kanssa. Lis\u00e4ksi funktionaalin normi\non yht\u00e4suuri Hilbertin avaruudessa m\u00e4\u00e4ritellyn yksik\u00e4sitteisen alkion normin\nkanssa.\nTuloksen esitt\u00e4mist\u00e4 varten tutkielmassa k\u00e4sitell\u00e4\u00e4n muun muassa metrisi\u00e4\navaruuksia sek\u00e4 tutustutaan lineaarisiin operaattoreihin. Ty\u00f6ss\u00e4 tarkastellaan\nmy\u00f6s normiavaruuksia, Lebesgue-avaruuksia, sis\u00e4tuloavaruuksia ja esitell\u00e4\u00e4n Hilbertin avaruus. Lis\u00e4ksi tutustutaan yleiseen approksimaatioongelmaan, jonka idea perustuu kompleksiarvoisten minimoivien alkioiden l\u00f6yt\u00e4miseen.\nRieszin esityslauseen sovelluksena tutustutaan kalvon mekaniikkaan. N\u00e4ytet\u00e4\u00e4n, ett\u00e4 kiinnitetyn v\u00e4r\u00e4htelev\u00e4n kalvon ominaisarvo-ongelma voidaan\nesitt\u00e4\u00e4 yksinkertaistetussa muodossa jatkuvan lineaarisen funktionaalin avulla, jonka j\u00e4lkeen sovelletaan Banachin kiintopistemenetelm\u00e4\u00e4 ominaisarvojen tulkintaan. Tutkielmassa osoitetaan, ett\u00e4 vaikka Rieszin esityslauseen avulla ei saada konkreettista lukuarvoa yht\u00e4l\u00f6n ratkaisulle, niin ratkaisun luonnetta pystyt\u00e4\u00e4n kuitenkin tarkastelemaan yksinkertaistetun muodon ansiosta.", "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 2025-02-19T09:45:04Z\nNo. of bitstreams: 0", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Made available in DSpace on 2025-02-19T09:45:04Z (GMT). No. of bitstreams: 0\n Previous issue date: 2023", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.format.extent", "value": "47", "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.title", "value": "Rieszin esityslause ja sen sovelluksia kiinnitetyn kalvon mekaniikassa", "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-202502192058", "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": "Matematiikan opettajankoulutus", "language": "fi", "element": "subject", "qualifier": "discipline", "schema": "dc"}, {"key": "dc.subject.discipline", "value": "Teacher education programme in 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.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": "Hilbertin 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"}]
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