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## Ma110c

**Analysis I (Spring 2014)**, instructor: Chris Marx
**Preliminaries** (probably from Lectures 1-4 or Ma110a)

The notes from Ma191b S7 are also useful.

1. Distributions and layer cake, Fubini-Tonelli

2. Riesz, Radon-Nikodym, Lebesgue decomposition (from Ma191b S7 3:Lp spaces)

3. Maximal inequalities (from Ma191b S7)

4. Summability kernels, convolutions, and approximate identities

5. Weak Lp spaces and weak-type operators

**Lecture notes**

Scanned lecture notes [bookmarked]

LaTeX'ed notes:

[I. HARMONIC ANALYSIS]

Lectures 1-4 (maximal inequalities most likely) are missing since I was attending Ma108c...

Lecture 5 (Real Interpolation, Marcinkiewicz interpolation, Radon-Nikodym decomposition)

Lecture 6 (Comparison of measures, de Valee-Poussin, Lebesgue decomposition theorem, Borel transform)

Lecture 7 (Applications to Fourier series)

Lecture 8 (Complex interpolation - Riesz-Thorin) [NEEDS PROOF]

Lecture 9 (Riesz-Thorin, Applications to Fourier series and convolutions - Hausdorff-Young, Young)

Lecture 10 (Convolutions, Young's Inequality, a.e. convergence of Fourier series, Christ-Kiselev maximal inequality)

Lecture 11 (Christ-Kiselev continuum version)

Lecture 12 (Hardy spaces)

Lecture 13 (Hardy spaces cont.)

Lecture 14 (Riesz factorization, Jensen's Formula)

Lecture 15 (Boundary values for Nevanlinna functions, brothers Riesz theorem)

Lecture 16 (Nevanlinna functions and Herglotz representation theorem)

Lecture 17-18 (Harmonic extensions, conjugates)

[II. OPERATOR THEORY]

Lecture 19 (Hilbert transform concluding remarks, operator topologies)

Lecture 20 (Adjoints, spectrum)

Lecture 21 (Resolvent operator, spectral radius)

Lecture 22 (Self-adjoint operators and the spectrum, polar decomposition)

Lecture 23 (Polar decomposition continued, functional calculus motivation)

Lecture 24 (Continuous functional calculus, extension to bounded Borel functions)

Lecture 25 (Functional calculus, spectral projections, multiplication sp??? form of spectral theorem)

Lecture 26 (Multiplication form of spectral theorem cont., spectral resolution, pvm)

Lecture 27 (Spectral resolution, spectral theorem for commuting s.a. operators and normal operators, Lebesgue decomposition of spectrum)

Lecture 28 (Discrete Laplacian in d-dimensions, physical interpretation of spectrum)

Lecture 29 (Scattering states, RAGE theorems)

All lecture notes concatenated

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