Analysis I, II, III

Lecture Notes

I will post lecture notes here: lecture notes

Course content

Analysis I is mainly concerned with measure theory and integration.

We will cover some of the most fundamental theorems / theories in Analysis

  • Measures, Lebesgue Measure, Hausdorff measure, Radon measures
  • Measurable and non-measurable sets and functions
  • Integration and Lp-spaces, Sobolev spaces
  • Lusin and Egorov
  • Convergence (in measure, almost everywhere, in Lp)
  • Fubini’s theorem
  • convolution and approximation
  • Radon-Nikodym
  • Riesz Representation
  • Lebesgue differentiation theorem
  • Area Formula and Transformation rule
  • Fourier transform and applications

Analysis II is focused on Sobolev spaces (including the necessary tools from Functional Analysis)

  • Dual spaces/Hahn Banach/Reflexive spaces and Weak convergence
  • Different formulations of Sobolev spaces
  • Trace theory
  • Embedding theorems

Analysis III treats Functional Analysis (with a focus on applications to Partial Differential Equations)

  • Fourier Transform
  • Topological Fixed Point Theorems
  • Hilbert Spaces
  • Open Mapping, Inverse Mapping, Closed Graph
  • Closed Range Theorem, Spectral Theory, Fredholm Alternative

Office hours

Office hours available upon request by email:

Or drop by my office. If I am there, and if I have time, I will probably talk to you.