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.