Analysis

• Lebesgue measure: Construction of Lebesgue measure on R and R^d. Measurable and non measurable sets; Cantor sets. Lebesgue and Borel measurable functions; Egorov theorem and Lusin Theorems. Construction and properties of Lebesgue integral; the space L¹ of integrable functions and its completeness; comparison with Riemann integral. Fubini-Tonelli theorem in R^d. Modes of convergence: convergence almost everywhere, convergence in measure, convergence in L¹.
• Integration and differentiation: Differentiation of the integral; Hardy-Littlewood maximal function; Lebesgue differentiation theorem. Functions of bounded variation; absolutely continuous functions; the fundamental theorem of calculus.
• Hilbert spaces Abstract Hilbert spaces and examples; L² spaces; Bessel’s inequality and Parseval’s identity; Riemann-Lebesgue Lemma; Orthogonality; orthogonal projections. Linear transformations; linear functionals; Riesz representation theorem; adjoint transformations.
• Fourier analysis: Fourier transform in L¹ and L²; Fourier inversion formula. Fourier series; Dirichlet’s Theorem and Fejér’s Theorem
• General theory of measure and integration Measure spaces and σ-algebras. σ-finite measures. Caratheodory theorem and the construction of measures; outer measures and extension theorems. Integration theory. Product measures and Fubini-Tonelli theorem. Signed measure; Radon-Nikodym theorem; Borel measures on R and Lebesgue-Stieljes integral.
• Banach spaces and L^p spaces. Abstract Banach spaces and examples; completeness criterion. Convexity; L^p-norms; Schwarz, Hölder, Minkowski, and Jensen inequalities. L^p-spaces and their duals; Riesz-Fischer Theorem.

References

• Berberian, Introduction to Hilbert Spaces
• Folland, Real Analysis
• Gelbaum and Olmsted, Counterexamples in Analysis
• Halmos, Measure Theory
• Royden, Real Analysis
• Rudin, Principles of Mathematical Analysis
• Rudin, Real and Complex Analysis
• Stein and Shakarchi, Real Analysis
• Wheeden and Zygmund, Measure and Integral