Syllabus for the Analysis Qualifying exam
 Real and Complex Analysis, Rudin
 Functional Analysis, Rudin
 Functional Analysis, Soboloev Spaces and Partial Differential Equations, Brezis
 Complex Analysis, Stein, Shakarchi
 Real Analysis for Graduate Students: Measure and Integration Theory, Bass
 Real Analysis, Stein, Shakarchi
 Fourier Analysis: An Introduction, Stein, Shakarchi
 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.