Brief review of Lebesgue measure on the real line; comparison of Riemann integration and Lebesgue
integration. Abstract measure and abstract integration, Monotone convergence theorem, Dominated
convergence theorem (DCT), Fatou’s lemma. Definition of the Fourier transform and Riemann-Lebesgue
lemma as a corollary of DCT.
Product sigma algebras, Product measures, Sections of measurable functions, Fubini’s theorem, Signed
measures and Radon-Nikodym theorem, Lp-spaces, Characterization of continuous linear functionals on
Lp - spaces, Change of variables, Complex measures, Riesz representation theorem.
Review of rudimentary functional analysis: Bounded linear functionals and dual spaces of Normed linear
spaces; the Hahn-Banach theorem. Bounded linear operators, open-mapping theorem, closed graph
theorem. Uniform Boundedness principle and applications to the existence of continuous functions on
the circle, whose Fourier series fails to converge at any prescribed point.
Review of Hilbert spaces, Riesz representation theorem, orthogonal complements. Reproducing kernel
Hilbert spaces as special examples of Hilbert spaces. Bounded operators on a Hilbert space up to (and
including) the spectral theorem for compact, self-adjoint operators.