1  Functional Analysis

The material for this section will mainly be taken from Rudin's [1,2]. Additional examples will be handed out as problem sheets.

Hilbert spaces.     Definition, Orthonormal bases, Riesz representation theorem H^* @ H. l²(\mathbbN) and L²(W).

Banach spaces.     Definition. A list of examples of function spaces which are Banach spaces: l^p(\mathbbN), L^p(W), C⁰(W), C^0,a(W), W^m,p(W), W₀^{m, p}(W), W^{-m, p}(W), BV(W). A function space which is not a Banach space: S(\mathbbRⁿ).

Banach spaces which are (not necessarily) function spaces: the dual X^*, the space of bounded operators L(X, Y) with operator norm.

Bounded operators.     Definition, some examples: differential operators P(D):C^mÆ C⁰, integral operators of various types (Young's inequality, infinite matrices on l^p, Hilbert-Schmidt operators).

The (Banach-) contraction mapping principle, ``Neumann-series" (a.k.a. the geometric series for (I+K)^-1).

General facts.     Hahn-Banach, Open Mapping and Closed Graph theorems. Closed subspaces in general do not have closed complements.

Adjoint operators, kerT^* = (rangeT)^{^}, kerT = ^{^}(rangeT).

Weak convergence. Definition, Banach-Alaoglu theorem. Riemann-Lebesgue lemma.

Compact operators and Fredholm operators.

2  Fourier series and integral

Fourier series on \mathbbTⁿ.     Uniform convergence of partial sums s_N(f) if f is Hölder continuous, convergence in L²(\mathbbTⁿ) for f ‘ L².

Fourier integral.     Inversion formula for f ‘ C¹_c(\mathbbRⁿ), pointwise for f with f, [^f] ‘ L¹(\mathbbRⁿ) and in the sense of distributions for f ‘ S¢(\mathbbRⁿ).

The Plancherel formula and boundedness of the Fourier transform on L²(\mathbbRⁿ).

Convolution and Fourier multipliers.     Boundedness on L², Hölder spaces (example of dyadic decomposition - notes will be provided) and without proof L^p.

Elliptic estimates for constant coefficient operators.

3  Sobolevology

Definition of the spaces W^{m, p}(W) with m ‘ \mathbbZ and of W₀^{m, p}(W) for any open subset W Ã \mathbbRⁿ. Invariance under diffeomorphisms (coordinate changes) of W.

Trace theorems.     Restriction of f ‘ W^{m, p}(W) to a smooth submanifold makes sense for suitable values of m and p. In particular one can restrict any f ‘ W^{1, p}(W) to �W if �W is smooth, and if p > n then one can ``restrict to a point,'' i.e. any f ‘ f ‘ W^{1, p}(W) is defined everywhere. In fact W^{1, p}(W) Ã C^{0, 1-n/p}(W).

Sobolev inequalities.     ||f||_L^n/(n-1) £ C ||‹f||_L¹, and consequences for other values of p than p = 1. Relation between the Sobolev inequality and the isoperimetric inequality.

Compactness issues.     The Rellich-Kondrachov compactness theorem (and Ascoli-Arzela of course).

Sobolev-like spaces.    (if time permits) BV(W) and sets with finite perimeter. Quick solution to the problem of minimal surfaces.

Solving PDEs by Hilbert-Sobolev space methods.     Solution of the Dirichlet problem by minimizing the Dirichlet integral. Solution of Poisson's equation by using the Riesz representation theorem. A proof of the Riemann Mapping Theorem from complex analysis, using real variable methods (if time permits).

References

[1]

W.Rudin, Real and Complex Analysis.

[2]

W.Rudin, Functional Analyis

[3]

L.C.Evans, Partial Differential Equations, A.M.S. Graduate Studies in Mathematics, 19 1998.