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A basic course in functional analysis focusing on applications of general theory in the context of the theory of partial
differential equations.
Last update: Šmíd Dalibor, Mgr., Ph.D. (11.06.2021)
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A. Bressan, Lecture notes on functional analysis: with applications to linear partial differential equations, American Mathematical Society, Providence, 2013
Ph. Ciarlet, Linear and nonlinear functional analysis with applications. SIAM, Philadelphia, 2013
A.N. Kolmogorov, S.V. Fomin, Elements of the theory of functions and functional nalysis, Dover publications, 1999 Last update: Málek Josef, prof. RNDr., CSc., DSc. (18.01.2022)
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An introductory course to functional analysis focused on the extension of the results of linear algebra to infinite-dimensional spaces and on applications of general theoretical results within partial differential equations.
1. Introduction
Finite-dimensional vector spaces and linear representations (summary). Function spaces, metric spaces, normed spaces. Banach and Hilbert spaces. Compactness in finite-dimensional and infinite-dimensional spaces.
2. Linear operators
Continuous linear operators, examples. Hahn-Banach theorem and its consequences. Dual spaces, weak and weak-* convergence. Reflexive spaces. Banach-Alaoglu theorem.
3. Bounded linear operators
Principle of uniform boundedness, open mapping theorem and closed graph theorem. Adjoint operator, compact operator.
4. Hilbert spaces
Orthogonal projections, Riesz representation theorem. Lax-Milgram lemma and its application in the theory of partial differential equations. Introduction to Sobolev spaces. Compact operators. Fredholm alternative. Spectrum. Self-adjoint operators, Hilbert-Schmidt theorem. Last update: Málek Josef, prof. RNDr., CSc., DSc. (14.01.2022)
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