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Basic course of linear functional analysis for bachelor's program in General Mathematics.
Recommended for specializations Mathematical Analysis and Mathematical Modelling and Numerical Analysis..
Last update: G_M (16.05.2012)
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The course is taught in Czech, so see the Czech version. Last update: Kalenda Ondřej, prof. RNDr., Ph.D., DSc. (12.09.2022)
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Habala, Hájek, Zizler, Banach Spaces I, II (skripta, MATFYZpress 1997) M. Katětov a J. Jelínek, Úvod do funkcionální analýzy (skripta, SPN Praha 1968) J. Lukeš, Uvod do funkcionální analýzy (skripta, Karolinum Praha, 2005) J. Lukeš, Zápisky z funkcionální analýzy (skripta, Karolinum Praha 1998, 2002, 2003) J. Lukeš a J. Malý, Míra a integrál (skripta, Univerzita Karlova, 1993, 2002 - anglické vydání 1995, 2005) L. Mišík, Funkcionálna analýza (Alfa Bratislava, 1989) K. Najzar, Funkcionální analýza (skripta, SPN Praha 1988) I. Netuka a J. Veselý, Příklady z funkcionální analýzy (skripta MFF UK 1972) P. Quittner, Funkcionálna analýza v príkladoch (Veda, SAV Bratislava 1990) W. Rudin, Analýza v reálném a komplexním oboru (Academia Praha 1977, 2003) W. Rudin, Functional analysis (Mc Graw Hill 1991 - ruský překlad 1975) J. Stará, Příklady z matematické analýzy IV: Funkcionální analýza (skripta, SPN Praha 1975) A.E. Taylor, Úvod do funkcionální analýzy (Academia Praha 1973) Last update: Spurný Jiří, prof. RNDr., Ph.D., DSc. (07.09.2012)
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The course is taught in Czech, so see the Czech version. Last update: Kalenda Ondřej, prof. RNDr., Ph.D., DSc. (12.09.2022)
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1. Banach and Hilbert spaces
normed spaces, spaces with inner product, examples of Banach spaces continuous linear mappings - characterization, norm, space of operators convergence of series in Banach spaces Hilbert spaces - orthonormal systems, orthonormal basis, Riesz-Fischer etc. finite-dimensional vs infinite-dimensional spaces real spaces vs. complex spaces 2. Duality and Hahn-Banach theorem Hahn-Banach theorem and its consequences separation of convex sets canonical embedding into second dual and reflexive spaces representation of dual spaces to classical Banach spaces wead (and weak*) convergence of sequences (definition, comparision, examples, characterization in classical spaces) choice of weakly convergent subsequences in reflexive spaces (and weak*-converent subsequences in duals of separable spaces) 3. Operators on Banach spaces Principle of uniform boundedness, Banach-Steinhaus and consequences Open mapping theorem and Closed graph theorem Quotient, projection, complementability Dual operators, duality of subspaces and quotients Adjoint operators between Hilbert spaces Spectrum of operators Compact operators - definition, properties, structure of the spectrum Selfadjoint compact operators on Hilbert space 4. Fourier transformation Definition and properties of Fourier transformation on L_1 Schwartz space and Fourier transformation on it Inverse theorem Plancherel transformation on L_2
Last update: Kalenda Ondřej, prof. RNDr., Ph.D., DSc. (12.09.2022)
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The lecture requires previous fair knowledge from mathematical analysis (Mathematical analysis 1-3, metric spaces from Mathematical analysis 4), linear algebra (mainly vector spaces and linear mappings, with emphasis on infinite-dimensional spaces), and Theory of measure and integral.
Last update: Kalenda Ondřej, prof. RNDr., Ph.D., DSc. (12.09.2022)
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