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These studies are available for the following degree students:</p><ul><li>Bachelor's Degree Programme in Mathematics</li><li>Master's Degree Programme in Mathematics</li><li>Bachelor's Degree Programme in Mathematics (Subject Teacher)</li><li>Master's Degree Programme in Mathematics&nbsp;(Subject Teacher)</li><li>Bachelor's Degree Programme in Mathematics, Chemistry or Physics Subject Teacher Education and Primary Teacher Education (Specialication in Mathematics)</li><li>Master's Degree Programme in Mathematics, Chemistry or Physics Subject Teacher Education and Primary Teacher Education (Specialication in Mathematics)</li><li>Doctoral Programme in Mathematics and Statistics</li><li>Doctoral Programme in Mathematics and Science (Specialication in Mathematics)</li><br></ul>","fi":"<p>Tämä opintojakso on tarjolla Matematiikan syventävät opinnot -ristiinopiskeluverkostossa. Verkoston opinnot ovat tarjolla seuraaville opiskelijoille:</p><ul><li>Matematiikan kandidaattiohjelma</li><li>Matematiikan maisteriohjelma</li><li>Matematiikan aineenopettajien kandidaattiohjelma</li><li>Matematiikan aineenopettajien maisteriohjelma</li><li>Matematiikan, kemian tai fysiikan aineenopettajan ja luokanopettajan kandidaattiohjelma (matematiikan opintosuunta)</li><li>Matematiikan, kemian tai fysiikan aineenopettajan ja luokanopettajan maisteriiohjelma (matematiikan opintosuunta)</li><li>Matematiikan ja tilastotieteen tohtoriohjelma</li><li>Matemaattisten tieteiden ja luonnontieteiden tohtoriohjelma (matematiikan opintosuunta)</li><br></ul>"},"cooperationNetwork":{"abbreviation":"matematiikansyventavat","name":{"en":"Cross-institutional studies in advanced courses in mathematics and statistics","fi":"Matematiikan ja tilastotieteen syventävien kurssien ristiinopiskelu","sv":"Korsstudier i fördjupade kurser i matematik och statistik"}}}],"gradeScaleId":"sis-0-5","outcomes":{"en":"By the end of the course, a student should \r\n-know, understand and be able to apply foundational concepts within analysis and functional analysis; and\r\n-be able to study independently typical contemporary graduate-level literature within these areas.","fi":"Efter godkänd kurs förväntas studenterna\r\n-känna till, förstå, och kunna använda de viktigaste matematiska begreppen inom analysen och funktionalanalysen på ett ändamålsenligt sätt; och\r\n-klara av att läsa och förstå allmänt hållen modern forskningslitteratur inom dessa områden.","sv":"Efter godkänd kurs förväntas studenterna\r\n-känna till, förstå, och kunna använda de viktigaste matematiska begreppen inom analysen och funktionalanalysen på ett ändamålsenligt sätt; och\r\n-klara av att läsa och förstå allmänt hållen modern forskningslitteratur inom dessa områden."},"tweetText":null,"content":{"en":"The courses Advanced analysis 1-2 pertain to the formalization of mathematical structures that in some sense generalize R^n and hence exhibit some similar properties. Such structures lay the foundation of, e.g., modern real and functional analysis as well as probability theory.\r\n\r\nPart 2: Fixed-point theorems with applications. Compactness and precompactness; the Arzela-Ascoli theorem with applications. Orthogonal complements in Hilbert spaces, orthogonal sequences and Fourier series; the Frechet-Riesz theorem.","fi":"Kurserna Fördjupad analys 1-2 handlar om formaliseringen av matematiska strukturer som i någon mening generaliserar R^n och har således vissa samma egenskaper. Dessa strukturer lägger grunden t.ex. för modern reell- och funktionalanalys eller sannolikhetslära.\r\n\r\nDel 2: Fixpunktssatser med tillämpningar. Kompakthet och prekompakthet; Arzela-Ascoli-teoremet med tillämpningar. Ortogonala komplement på Hilbertrum; ortogonala följder och Fourierserier; Frechet-Riesz-teoremet.","sv":"Kurserna Fördjupad analys 1-2 handlar om formaliseringen av matematiska strukturer som i någon mening generaliserar R^n och har således vissa samma egenskaper. Dessa strukturer lägger grunden t.ex. för modern reell- och funktionalanalys eller sannolikhetslära.\r\n\r\nDel 2: Fixpunktssatser med tillämpningar. Kompakthet och prekompakthet; Arzela-Ascoli-teoremet med tillämpningar. Ortogonala komplement på Hilbertrum; ortogonala följder och Fourierserier; Frechet-Riesz-teoremet."},"additional":{"en":"The course is taught roughly every second year","sv":"Kursen undervisas ungefär vartannat år"},"prerequisites":{"en":"BSc level analysis and calculus.","sv":"Analys, del 1 och del 2."},"compulsoryFormalPrerequisites":[],"recommendedFormalPrerequisites":[],"literature":[],"learningMaterial":null,"completionMethods":[]}],"prerequisiteCourseUnit":[],"prerequisiteModule":[]},"prerequisiteCourseUnitPage":{"nodes":[]},"prerequisiteModulePage":{"nodes":[]},"parentModulePage":{"nodes":[]}},"pageContext":{"type":"courseUnit","locale":"en","title":"Advanced Analysis II","id":"otm-279bc530-b381-37d6-9aea-195a6ca63221","code":"273050.0","prerequisiteCourseUnitIds":[],"prerequisiteModuleIds":[],"parentModuleIds":[],"curriculumPeriodStartDate":"2026-08-01","curriculumPeriodEndDate":"2027-08-01","coordinatingOrgIds":[],"searchable":false,"searchTags":null,"organisationIds":["otm-5a6a3701-b188-3724-9476-784e433a4d92"],"organisations":[],"attainmentLanguages":[],"hasSummerStudies":false,"teachingPeriods":[],"cooperationNetworkDirection":"INBOUND","hasCooperationNetworkSettings":true,"hasAvoinTeaching":false}}}