# 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: armin@pitt.edu

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