Faculty of Mathematics and Physics

Functional Analysis 2

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


Content of the lectures and classes


Conditions for exams

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

Functional analysis 2 is an advanced course designed mainly for master students of mathematical analysis. This course is a continuation of the course Functional Analysis 1 (NMMA401) which itself is already an advanced master course and on the bachelor course Introduction to Functional Analysis (NMMA331).


Content of the course

The basic topic of the course is spectral theory for elements of Banach algebras and for operators on Hilbert spaces. The content is divided to four chapters:

  • Banach algebras and Gelfand transform
  • C*-algebras and continuous function calculus
  • Bounded and unbounded operators on a Hilbert space
  • Spectral measures and decompositions


The first topic: The theory of Banach algebras is an abstract theory covering algebras of bounded operators (and so generalizing Section III.6 from Introduction to Functional Analysis), algebras of continuous functions and convolution algebras. In addition to basic properties of spectra we will focus on the functional calculus (which enables us to plug elements of an algebra into holomorphic functions) and on the Gelfand transform (which provides a representation of some commutative algebras and simultaneously generalizes the Fourier transform and the Fourier series).


The second topic: C*-algebras form an important subclass of Banach algebras, which includes algebras of bounded (or compact) operators on Hilbert spaces and also algebras of continuous functions. We will focus on spectial properties of spectra and of homomorphism of C*-algebras and also on continuous functional calculus (which enables us to plug certain elements into continuous functions, using the Gelfand transform from the previous topic).


The third topic: We will focus on unbounded operators - on basic definitions and properties, on their spectra and on the notion of the adjoint operator. Unbounded operators are linear operators which need not be continuous nor defined on the whole space. Important examples include differential operators.


The fourth topic: We start by the measurable calculus for normal operators, which is a generalization of the continuous calculus from the second topic that enables us to plug normal operators to suitably measurable functions. The measurable calculus will be used to construct the spectral decomposition of a normal operator, which may be viewed as a generalization of the Hilbert-Schmidt theorem (from Section III.7 of Introduction to Functional Analysis) to non-compact operators. A generalization to some unbounded operators will follow. The last two sections are not assumed to be presented in detail, they are included mainly for possibly interested students.


Expected knowledge

For a meaningful study of such an advanced mathematical course a reasonable knowledge of many areas of mathematics is necessary. The most important ones are the following:

  • Elements of functional analysis (Banach and Hilbert spaces, bounded operators, basic theorems of functional analysis) - throughout all the course.
  • Elements of general topology and its application in functional analysis (compactness, weak* topology) - mainly in the first two topics.
  • Complex analysis (holomorphic functions, path integral, Cauchy theorem, Cauchy formula) - mainly in the first topic.
  • Measure and integral (measure, abstract Lebesgue integral, limit theorems, Borel measures on metric spaces) - mostly in the fourth topic, but also elsewhere.


How to continue?

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

  • Partial differential equations 1,2 (NMMA405, NMMA406), Diferential equations in Banach spaces (NMMA440) - applications of functional analysis to studying the solutions of equations
  • 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)