Analysis of Functions

Analysis of Functions(D)

ANALYSIS OF FUNCTIONS (D)
Part II Linear Analysis and Part II Probability and Measure are essential.

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 6= ∞. [3]

Banach and Hilbert space analysis

  • Strong, weak and weak-* topologies. Review of the Riesz representation theorem for Hilbert spaces; reflexive spaces. Orthogonal systems of functions and their completeness. Hermite polynomials, the Haar basis. Compactness: review of the Ascoli–Arzel`a theorem; weak-* compactnesss of the unit ball (both the separable and non-separable cases). The Riesz representation theorem for spaces of continuous functions. The Hahn–Banach theorem. Review of the Baire category theorem and its consequences: the open mapping theorem and the Banach–Steinhaus theorem. [5]

Fourier analysis

  • Definition of Fourier transform in L1. Extension to L2 by density and Plancherel’s isometry. Fourier inversion theorem. Duality between regularity in real variable and decay in Fourier variable. Representation of L2 periodic functions by Fourier series; the Poisson summation formula. Construction of solutions for linear PDEs with constant coefficients. [3]

Generalized derivatives and function spaces

  • Definition of generalized derivatives and of the basic spaces in the theory of distributions: D/D0 and S/S0. Definition of the Sobolev spaces Hk in Rd and the periodic d-cube. Sobolev embedding. The Rellich–Kondrashov theorem. The trace theorem. Construction and regularity of solutions for the Dirichlet problem of Laplace’s equation. [5]

Appropriate books

  • H. Br´ezis Functional Analysis, Sobolev Spaces and Partial Differential Equations. Universitext, Springer 2011
  • A.N. Kolmogorov, S.V. Fomin Elements of the Theory of Functions and Functional Analysis. Dover Books on Mathematics 1999
  • E.H. Lieb and M. Loss Analysis. Second edition, AMS 2001

Associated GitHub Page
https://jaircambridge.github.io/Analysis-of-Functions-D/