Analysis of Functions (Lent 2023)

Time and Location: Tu-Th-Sa 10am, MR4

Analysis of Functions is a course in Part II of the Cambridge Tripos.

Schedule

  • Lebesgue integration theory. Review of integration: simple functions, monotone and dominated convergence; existence of Lebesgue measure; definition of Lp spaces and their completeness. The Lebesgue differentiation theorem. Egorov’s theorem, Lusin’s theorem. Mollification by convolution, continuity of translation and separability of Lp when p ≠ ∞.
  • Banach and Hilbert space analysis. Strong, weak and weak-* topologies; reflexive spaces. Review of the Riesz representation theorem for Hilbert spaces; the Radon–Nikodym theorem; the dual of Lp. Compactness: review of the Ascoli–Arzela theorem; weak-* compactnesss of the unit ball for separable Banach spaces. The Riesz representation theorem for spaces of continuous functions. The Hahn–Banach theorem and its consequences: separation theorems; Mazur’s theorem.
  • Fourier analysis. Definition of Fourier transform in L1; the Riemann–Lebesgue lemma. Fourier inversion theorem. Extension to L2 by density and Plancherel’s isometry. Duality between regularity in real variable and decay in Fourier variable.
  • Generalized derivatives and function spaces. Definition of generalized derivatives and of the basic spaces in the theory of distributions: D/D′ and S/S′. The Fourier transform on S′. Periodic distributions; Fourier series; the Poisson summation formula. Definition of the Sobolev spaces Hs in Rd. Sobolev embedding. The Rellich–Kondrashov theorem. The trace theorem.
  • Applications. Construction and regularity of solutions for elliptic PDEs with constant coefficients on Rn. Construction and regularity of solutions for the Dirichlet problem of Laplace’s equation. The spectral theorem for the Laplacian on a bounded domain. *The direct method of the Calculus of Variations.*

Lecture notes (will be updated throughout the term)

Example Sheets

The example sheets will be posted here every two weeks, starting with the first week of term.

References

  • C. Warnick, Analysis of Functions, Lecture Notes. Available here.
  • E.H. Lieb and M. Loss, Analysis, American Mathematical Society.
  • G.B. Folland, Real Analysis, Wiley.