Applied Analysis by Hunter and Nachtergaele, Chapters 1, 3-6
http://www.math.ucdavis.edu/~hunter/book/pdfbook.html

Graduate standing in Mathematics or Applied Mathematics, or consent of instructor

Metric and normed spaces. Continuous functions. Topological, Hilbert, and Banach spaces. Fourier series. Spectrum of bounded and compact linear operators. Linear differential operators and Green's functions. Distributions. Fourier transform. Measure theory. Lp and Sobolev spaces. Differential calculus and variational methods.

 

Lectures	Sections	Topics/Comments
4		Review of analysis on metric spaces: Definitions and ex- amples of metric spaces, p-norms on Rn, Cauchy-Schwarz and Minkowski inequalities, continuity, open and closed sets, and completeness. Definition of topological spaces.
5		Compactness: Definitions and examples of sequential compactness and the open-set definition of compactness, equivalence of norms in finite-dimensional normed vector spaces, motivating examples of com-pactness in infinite-dimensional spaces, Riesz’s Lemma, and proof that the closed unit ball in an infinite-dimensional normed vector space is not compact.
5		Spaces of continuous functions: definition of C(K) for K compact, uniform convergence and completeness, definition and examples of equicontinuity, Arzel`a-Ascoli theorem, applications of the Arzel`a-Ascoli theorem, and Stone-Weierstrass theorem (proof optional or only the Weierstrass approximation theorem in 1D).
3		Banach spaces: Definition of Banach spaces, examples of Banach spaces (Cn(K), c, c0, lp, Lp), space of bounded and compact linear operators, dual spaces, Hahn-Banach theorem for extension of bounded linear functionals (proof optional).
6		Hilbert spaces: Definition of Hilbert spaces, examples of Hilbert spaces (l2, L2, W1,2) , orthogonality and projections, orthonormal bases, Riesz representation theorem, characterization of compact sets, and the approximation property of Hilbert spaces.
3		Weak convergence: Definitions of weak and weak* convergence, examples of weak versus strong convergence, definition and examples of weak sequential compactness, Banach-Alaoglu theorem (the closed unit ball is weakly sequentially compact in a reflexive Banach space; other versions optional), and applications.

 

Post lecture notes with exercises on basic 127-level material prior to start of lecture (both Hunter and Shkoller have this material available) TAs can review solutions to exercises during the first and second discussion section.