My friend Ethan Jaffe wrote some notes about various topics in Analysis during our time at MIT. You may find them below:

• Distribution Theory
• Fourier transform of homogeneous distributions
• Fourier transform of homogeneous radial distributions
• On Smoothing operators
• A proof of the Schwartz kernel theorem
• Functional Analysis
• Analytic Fredholm theory
• Atkinson's theorem
• Essential spectrum
• Fredholm determinant
• Karamata's inequality
• Microlocal Analysis
• Fredholm properties of elliptic operators
• Oscillatory integrals
• Sharp Gårding inequality
• Wavefront set and coordinate invariance
• Sobolev Spaces
• Counterexamples to Sobolev embedding and trace
• Kondrashov compactness
• Defining $H^s (\mathbb{T}^n)$
• Products in Sobolev spaces
• Miscellaneous
• Wave equation on manifolds
• Fundamental theorem of Algebra
• Inverse function theorem
• Jacobi's formula and the Laplace-Beltrami operator
• Principal Component Analysis