Syllabus for the Analysis Qualifying exam

Recommended books:

[1] Real and Complex Analysis, Rudin
[2] Functional Analysis, Rudin
[3] Functional Analysis, Soboloev Spaces and Partial Differential Equations, Brezis
[4] Complex Analysis, Stein, Shakarchi
[5] Real Analysis for Graduate Students: Measure and Integration Theory, Bass
[6] Real Analysis, Stein, Shakarchi
[7] Fourier Analysis: An Introduction, Stein, Shakarchi
[8] Functional Analysis, Stein, Shakarchi

Measure Theory [1,5 and 6]

Measures, Measure spaces, Measurable sets, Convergence theorems (Fatou’s, monotone and dominated), Lebesgue measure, Cauchy-Schwartz inequality, Parallelogram law, $L_p$ and $l^p$ spaces and norms, inequalities: Hölder, Minkowski and Jensen, Egorov’s Theorem, Fubini’s Theorem.

Functional Analysis [2,3,5 and 7]

Topological vector spaces, Linear operators: continuous, self-adjoint, compact, Baire Category Theorem, Banach Steinhaus Theorem, Open mapping Theorem, Closed Graph theorem, Hahn Banach Theorem, Banach Alaoglu Theorem, Krein-Milman Theorem, Extreme points in convex sets, Duality of Banach Spaces, Fixed Point Theorem (basic application to PDE), Spectral theorem for compact self-adjoint operators.

Harmonic Analysis [1,5,6 and 7]

Hilbert spaces, Banach Spaces, ONB, Fourier series, Fourier transform, Basic properties of Fourier transform and series, Plancherel Theorem, Parseval theorem, Convolution.

Complex Analysis [1 and 4]

Cauchy theorem, Residue calculus, Liouville theorem, Contour integrals, Rouché Theorem, Poisson integral, Blaschke’s factors, Conformal mappings, Maximum modulus principle, Holomorphic functions, Argument principle, Schwarz lemma, Riemann mapping theorem.