Workshop schedule
Week 1 (May 20): preliminary review
Uniform convergence, the Weierstrass \(M\)-test
Nowhere dense sets, Baire category, the Heine–Borel theorem
Normed spaces
Sequences and series, convergence, tails, \(\limsup\) and \(\liminf\), Cauchy criteria for sums and integrals
The triangle and Cauchy–Schwarz inequalities
The mean value theorem, Taylor’s theorem and its remainder
Weierstrass approximation
Pathological functions and continuity
Week 2 (May 27): measure theory for sets
\(F_\sigma\) and \(G_\delta\) sets
\(\sigma\)-algebras
Outer measure, Lebesgue measure
Continuity of measure
The Borel–Cantelli lemma
Week 3 (June 3): integration I
Measurable functions
Lebesgue integration, change of variables
Chebyshev’s inequality
Egorov’s and Lusin’s theorems
Modes of convergence: uniform, pointwise, almost everywhere, in measure, in norm
Week 4 (June 10): integration II
Fatou’s lemma, the monotone and dominated convergence theorems
The Fubini–Tonelli theorems and iterated integration
Week 5 (June 17): Fourier analysis
The Fourier transform
Fourier series
Convolutions, approximations to the identity
Trigonometric series, density of trigonometric polynomials
Week 6 (June 24): functional analysis I
Banach and Hilbert spaces, completeness
\(\ell^p\) and \(L^p\)
Hölder’s inequality
Dual spaces, linear functionals, the operator norm
Week 7 (July 1): functional analysis II
\(L^1\), \(L^2\), \(L^\infty\), the Cauchy–Schwarz inequality, the Pythagorean theorem for orthogonal vectors
Function spaces
Plancherel’s theorem, Parseval’s identity, Bessel’s inequality, the Riesz representation theorem
Week 8 (July 8): abstract measures
- The Radon–Nikodym theorem
Topics from other real analysis syllabi
Convergence theorems for integrals, Borel measures, the Riesz representation theorem
\(L^p\) spaces and their duals, Jensen’s inequality
The Lebesgue differentiation theorem, Fubini’s theorem, Hilbert spaces
Complex measures of bounded variation, the Radon–Nikodym theorem
Fourier series, the Fourier transform, convolution
The heat equation, the Dirichlet problem, fundamental solutions
The central limit theorem, the law of large numbers, conditional probability and conditional expectation
Distributions, the Sobolev embedding theorem
The maximum principle
Measure theory
Outer measure, measure, measurability, \(\sigma\)-algebras, Lebesgue measure on \(\RR^n\), Borel and Radon measures, approximation by open and closed sets.
Measurable functions: measurability of \(\inf\), \(\sup\), \(\liminf\), \(\limsup\) of sequences and of algebraic operations, simple functions, Lusin’s and Egorov’s theorems, convergence in measure and in probability.
Integration: the bounded convergence theorem, Fatou’s lemma, the monotone convergence theorem, absolute continuity of the integral.
Differentiation and integration: the Vitali covering lemma, almost everywhere differentiability of monotone functions, the integral of the derivative of a monotone function, functions of bounded variation, differentiation of the indefinite integral, the fundamental theorem of calculus for absolutely continuous functions.
Product measures: the Fubini and Tonelli theorems.
Differentiation of measures: absolute continuity of measures, the Radon–Nikodym theorem, mutually singular measures, the Lebesgue decomposition, Lebesgue points, the Lebesgue–Besicovitch theorem, signed measures.
The Riesz representation theorem on \(L^p\) for \(1\leq p<\infty\), the Riesz–Markov theorem on the dual of \(C_c\), the Riesz–Thorin interpolation theorem.
Functional analysis
Hilbert spaces: direct sums, orthogonal complements, the Riesz representation theorem, orthonormal bases in the separable and non-separable cases, the Fourier basis for \(L^2(\mathbb T)\), \(\mathbb T = \RR/2\pi\ZZ\).
Banach spaces: continuous linear maps, duals, direct sums and quotients, the Hahn–Banach theorem, the Baire category theorem, the Banach–Steinhaus theorem (uniform boundedness principle), the open mapping theorem.
Topological spaces: bases and local bases, convergence, continuity, compactness, weak topologies, the weak and weak-\(*\) topologies on Banach spaces, the Stone–Weierstrass theorem, the Banach–Alaoglu theorem, the Riesz–Markov theorem.
Locally convex spaces: the Minkowski gauge, the equivalence of convex balanced absorbing neighborhoods and seminorms, the Hahn–Banach theorem, metrizability, Fréchet spaces, the Banach–Steinhaus and open mapping theorems.
Distributions: \(C^\infty(\mathbb T)\) and its dual \(\mathcal D'(\mathbb T)\), Schwartz functions \(\mathcal S(\RR^n)\) and tempered distributions \(\mathcal S'(\RR^n)\), differentiation of distributions, multiplication of distributions by smooth functions, convergence of sequences of distributions, the distributions \((x\pm i0)^{-1}\) and \(\delta_0\), support of distributions.
Bounded operators on Banach spaces: adjoints, the spectrum, the resolvent, the spectrum of bounded self-adjoint operators on Hilbert spaces, integration and differentiation with values in Banach spaces, the spectral radius, continuous functional calculus for bounded self-adjoint operators on Hilbert spaces and the spectral measure.
Compact operators on Banach spaces: norm closure, composition and adjoint properties, approximation by finite-rank operators in separable Hilbert spaces, the analytic Fredholm theorem, the Riesz–Schauder and Hilbert–Schmidt theorems.
Fourier analysis
Fourier series on \(\mathbb T = \RR/2\pi\ZZ\): orthogonality, \(L^2\) completeness, uniform convergence and its failure, Fourier series in \(C^\infty(\mathbb T)\) and \(\mathcal D'(\mathbb T)\), interaction with differentiation and multiplication by \(e^{i\theta}\), the Sobolev spaces \(H^s(\mathbb T)\), compactness of the inclusion \(H^s(\mathbb T)\subseteq H^r(\mathbb T)\) for \(s>r\), the representation of distributions as derivatives of continuous functions, the Schwartz kernel theorem.
The Fourier transform on \(\RR^n\): isomorphism on \(\mathcal S(\RR^n)\) and \(\mathcal S'(\RR^n)\), isometry on \(L^2(\RR^n)\), interaction with differentiation and multiplication by the coordinate functions \(x_j\), the Fourier transform of compactly supported distributions, the Fourier transform of Gaussians and the heat equation, the Sobolev spaces \(H^s(\RR^n)\) and the inclusion \(H^s(\RR^n)\subseteq C^0(\RR^n)\) for \(s>n/2\).