Ondrej Kalenda

Functional Analysis 1

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Organization of the course

Content of the course, expected knowledge
and connections to other courses

Recommended literature

Lecture notes

Credit requirements
(and statistical data on homeworks)

Content of the lectures and classes

Conditions for exams (examination has been finished)

Overview of exam results

Content of the course, expected knowledge and connections to other courses

Functional analysis 1 is an advanced course for master students of mathematical analysis. Therefore the knowledge on the level of the bachelor program General mathematics, specialization Mathematical analysis is expected.

More specifically, this course is a kind of continuation of the bachelor course Introduction to functional analysis (NMMA331). Besides, we will need a basic knowledge of general topology taught in the bachelor course General topology 1 (NMMA345, formerly NMMA335) and a sound knowledge of measure and integration.

Basic topics of the course are the following:

Locally convex spaces and weak topologies Elements of the theory of distributions Elements of vector integration Compact convex sets

The knowledge covered by Introduction to functional analysis will be used throughout the course. To understand the first topic one moreover needs to know basic notions and results from general topology. Some of them will be briefly recalled, but there is no time for a detailed exposition. The necessary knowledge is summarized in the appendix on general topology which forms a part of the lecture notes. In the second topic we will use the Lebesgue integral on Rⁿ and also the diffential calculus of several variables. The third topic is devoted to a generalization of the Lebesgue integral to the case of vector-valued functions, therefore one needs to know measure theory and abstract Lebesgue integration. The fourth topic uses some notions from the first and third ones.

How to continue?

There are many further courses devoted to functional analysis and its applications, e.g.:

Functional analysis 2 (NMMA402) - a continuation of this course devoted to the spectral theory in Banach algebras and for operators on Banach (and mainly Hilbert) spaces. Partial differential equations 1,2 (NMMA405, NMMA406) - applications of functional analysis to studying the solutions of equations, it uses knowledge from the Introduction to functional analysis and, among others, from the second and the third areas of this course. Diferential equations in Banach spaces (NMMA440) Topological methods in functional analysis 1,2 (NMMA435, NMMA436) - a deeper study of weak topologies and of differentiability of convex functions on Banach spaces Introduction to the theory of approximations 1,2 (NMMA565, NMMA566) - applications of functional analysis to the study of approximations, i.e., of the nearest points Introduction to the theory of interpolations 1,2 (NMMA533, NMMA534) - applications of functional analysis to the study of various function spaces Nonlinear functional analysis 1, 2 (NMMA501, NMMA502)