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Course, academic year 2023/2024
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Functional analysis for physicist - NMMO302
Title: Funkcionální analýza pro fyziky
Guaranteed by: Mathematical Institute of Charles University (32-MUUK)
Faculty: Faculty of Mathematics and Physics
Actual: from 2022 to 2023
Semester: summer
E-Credits: 8
Hours per week, examination: summer s.:4/2, C+Ex [HT]
Capacity: unlimited
Min. number of students: unlimited
4EU+: no
Virtual mobility / capacity: no
State of the course: taught
Language: English, Czech
Teaching methods: full-time
Teaching methods: full-time
Guarantor: prof. RNDr. Josef Málek, CSc., DSc.
Class: M Bc. MOD
M Bc. MOD > Povinné
M Bc. OM
M Bc. OM > Povinně volitelné
Classification: Mathematics > Mathematics General, Mathematical Modeling in Physics
Annotation -
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)
Literature -

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)
Syllabus -

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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