Mathematics

Overview > Analysis

Fundamental for analysis is the theory of convergence – therefore, it covers (the convergence of) sequences and series, as well as differential and integral calculus. When studying mathematics at the Vienna University of Technology, one learns about these basics in the three courses "Analysis I – III" during the first three terms:

Furthermore, there are several other branches which are part of analysis, too, such as the Lebesgue measure and integration theory, which is covered by the course "Measure Theory". In this course, one deals with measures – these are functions which assign a non-negative real number (e.g. length, area, volume, etc.) to certain sets, whereby the measure of the union of all sets in a countable set of sets equals the sum of the measure of all individual sets – and with integrals which are defined with the help of such measures. So, one eventually arrives at a slightly more general definition for integrals than the one using Riemann sums.

Moreover, analysis deals with Differential Equations which are certain equations that establish a relationship between the unknown functions and their derivatives. In this connection, one usually wants to determine the functions defined by these equations (which is only possible in rather rare cases) or to determine at least some properties of these functions (periodicity, boundedness, etc.).

Funktional Analysis is a part of analysis, too: Based on topology, one studies the basics of metric spaces, Banach and Hilbert spaces (together with some important appropriate theorems such as the Hahn-Banach theorem or the Riesz representation theorem). In addition, it is dealt with spectral theory of linear operators (among others).

Finally, the last part of this branch is Complex Analysis which deals with complex funktions. However, some results may also be used for solving problems in the real numbers, for example concerning certain integrals.