Syllabus
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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
Annotation
An introductory course in functional analysis. Not equivalent to the course NMMA331.