Faculty of Mathematics and Physics

Functional Analysis 1

Information in Student Information System


Organization of the course


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


Recommended literature


Lecture notes


Credit requirements (including overview of results)


Content of the lectures and classes


Conditions from exams


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 complex analysis (NMMA301). Besides, we will need a basic knowledge of general topology taught in the bachelor course General topology 1 (NMMA335), a sound knowledge of measure and integration and several things from complex analysis.


Basic topics of the course are the following:

  • Topological vector spaces and weak topologies
  • Elements of vector integration
  • Banach algebras, operators on a Hilbert space and spectral theory


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 summed up in the appendix on general topology which forms a part of the lecture notes. The second 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. Within the third topic we will use, among otheres, some facts from complex analysis (properties of holomorphic functions) and also from measure theory and integration.


How to continue?

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

  • Functional analysis 2 (NMMA402) - a direct continuation of this course, advanced topics from the first and third areas
  • 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 area of this course
  • Diferential equations in Banach spaces (NMMA440) - some notions and results from the third area and their generalizations are used
  • 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)