Notes Basic Exam: Here are the notes I wrote in preparation for the basic exam. Analysis Qualifying Exam: Here is a semi-comprehensive set of notes for the UCLA analysis qualifying exam. Analysis Problem Topics: Select Tag a.e. convergence (1) absolute continuity (3) adjoint (2) approximation to the identity (2) argument principle (2) Arzela-Ascoli theorem (3) Baire category theorem (4) Banach spaces (2) Banach-Alaoglu theorem (1) Blaschke factors (2) Blaschke product (1) Cantor function (1) Cauchy-Schwarz (3) change of variables (2) Chebyshev’s inequality (3) closed graph theorem (1) compact operators (2) complex analysis (14) conformal map (3) convergence in measure (1) convex analysis (1) countable (1) density argument (8) dyadic decomposition (1) Eberlein-Smulian theorem (1) Egorov’s theorem (1) fat Cantor set (1) fixed points (1) Fourier series (6) Fourier transform (5) Fourier transform of measure (1) Fredholm operators (1) Fubini-Tonelli (2) functional analysis (8) Great Picard’s theorem (2) Green’s theorem (1) Hadamard factorization theorem (2) Hahn-Banach theorem (3) Hardy-Littlewood maximal function (1) harmonic analysis (1) harmonic functions (1) Hilbert spaces (1) Hölder’s inequality (4) infinite products (1) integrability (1) Jensen’s formula (2) Jensen’s inequality (1) layer cake decomposition (2) Lebesgue differentiation theorem (3) Liouville’s theorem (4) Lipschitz continuity (1) Little Picard’s theorem (2) Lp spaces (5) Lusin’s theorem (1) Marcinkiewicz interpolation (1) maximal function (1) maximum modulus principle (4) measurability (2) measure theory (4) meromorphic function (1) Minkowski inequality (1) Mittag-Leffler’s theorem (1) Möbius transformations (2) monotone functions (1) Montel’s theorem (1) Morera’s theorem (1) normal families (1) open mapping theorem (1) Plancherel’s theorem (2) portmanteau lemma (1) power series (1) Prokhorov’s theorem (1) Radon-Nikodym theorem (2) regularity of measure (1) removable singularity (1) residue theorem (1) Riemann-Lebesgue lemma (1) Riesz representation theorem (4) Riesz-Markov-Kakutani theorem (1) Riesz-Thorin interpolation theorem (1) Rouche’s theorem (1) Schwartz functions (1) Schwarz lemma (2) Schwarz lemma (1) Schwarz reflection principle (2) second-countable (3) seminorms (1) separability (2) sigma algebras (2) smooth functions (1) Sobolev spaces (2) spectral theory (1) Steinhaus lemma (1) Stone-Weierstrass theorem (1) subharmonic functions (3) support of measure (1) topology (2) translation operators (1) uniform boundedness principle (1) Vitali convergence theorem (2) weak convergence (8) weak topology (1) weak-* convergence of measure (1) weak-* topology (4) Analysis Problem of the Day 101 Analysis Problem of the Day 100 Analysis Problem of the Day 99 Analysis Problem of the Day 98 Analysis Problem of the Day 97