Exam Overview
- Exam Focus: Real Analysis and Functional Analysis
- Exam Format: 4-Hour Written Exam
- View Past Analysis Exams »
Syllabus Topics
This exam will test your working knowledge of basic real and functional analysis. You will be required to demonstrate an ability to use standard results and techniques to solve problems, including special cases of standard theorems which do not require long arguments.
We will not emphasize the memorization of statements of theorems nor of long proofs of standard theorems. The student is urged to work on the problems in the relevant sections and chapters in the reference books.
Syllabus updated Summer 2026. There will typically be 10-12 problems covering both Real Analysis and Functional Analysis. Students must designate 10 to be graded.
Real Analysis
- Abstract measure spaces and construction of measures.
- Measurable functions and general theory of integration.
- Integral Convergence Theorems (Fatou, Monotone, Dominated, Vitali).
- Modes of convergence: convergence a.e.; convergence in measure and convergence in the mean and how they are related to each other, Egorov's Theorem, Luzin's Theorem.
- Absolute continuity and the Radon-Nikodym Theorem.
- Product Measures and Fubini-Tonelli Theorem.
- Lebesgue Differentiation Theorem; Comparison of Riemann and Lebesgue integrals; Bounded Variation etc.
- Lp spaces with 1≤p≤∞, Hölder inequality, Minkowski inequality, Jensen's inequality, convergence in Lp, completeness of spaces, dense sets of Lp( a,b) (1≤p<∞).
Functional Analysis
- Metric spaces, Baire Category Theorem, Stone-Weierstrass Theorem.
- Normed linear spaces.
- Completeness of Lp spaces, dual spaces, Riesz Representation Theorem.
- Basic principles in Banach spaces (convex sets, Hahn-Banach, extreme points, Krein-Milman).
- Elements of Hilbert space theory (orthonormal basis, Riesz, Lax-Milgram, Bessel, Parseval).
- Fourier expansions (Riemann-Lebesgue, convergence), Fourier transforms (convolutions, application to Central Limit Theorem).
- Bounded linear operators (norms, adjoint, Hermitian, Bounded-inverse Theorem).
- Spectral theorem (resolvent, spectrum, eigenvalues) for compact operators, finite-rank operators, Fredholm Alternative.
References
- Natanson, Theory of Functions of a Real Variable.
- Kolmogorov and Fomin, Introductory Real Analysis.
- Royden, Real Analysis.
- Rudin, Real and Complex Analysis, 2nd edition.
- Friedman, Foundations of Modern Analysis.
- Folland, Real Analysis.
- Wheeden and Zygmund, Measure and Integral, an introduction to real analysis.
- Stein and Sharkarchi, Real Analysis, Measure Theory, Integration and Hilbert Spaces.
- Stein and Sharkarchi, Fourier Analysis.
