Strong emphasis on function spaces and structures in analysis. Detailed proofs of theorems.
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| \(C[0,1]\): the Faber-Schauder basis, the Riesz representation theorem, and the Borel \(\sigma\)-algebra | PDF |
| \(C^k\) spaces and spaces of test functions | PDF |
| \(L^0\), convergence in measure, equi-integrability, the Vitali convergence theorem, and the de la Vallée-Poussin criterion | PDF |
| \(L^1\) norms of products of sines | PDF |
| \(L^p\) norms of a sine sum | PDF |
| \(L^p\) norms of trigonometric polynomials | PDF |
| A series of secants | PDF |
| Abstract Fourier series and Parseval's identity | PDF |
| Alternating multilinear forms | PDF |
| Banach algebras | PDF |
| Bernoulli polynomials | PDF |
| Chebyshev polynomials | PDF |
| Compact operators on Banach spaces | PDF |
| Convolution semigroups, canonical processes, and Brownian motion | PDF |
| Cyclotomic polynomials | PDF |
| Decomposition of the spectrum of a bounded linear operator | PDF |
| Denjoy's theorem on circle diffeomorphisms | PDF |
| Diophantine vectors | PDF |
| Explicit construction of the \(p\)-adic numbers | PDF |
| Fatou's theorem, Bergman spaces, and Hardy spaces on the circle | PDF |
| Fréchet derivatives and Gâteaux derivatives | PDF |
| Functions of bounded variation and a theorem of Khinchin | PDF |
| Functions of bounded variation and differentiability | PDF |
| Gaussian integrals | PDF |
| Gaussian measures and Bochner's theorem | PDF |
| Gaussian measures, Hermite polynomials, and the Ornstein-Uhlenbeck semigroup | PDF |
| Germs of smooth functions | PDF |
| Gibbs measures and the Ising model | PDF |
| Gradients and Hessians in Hilbert spaces | PDF |
| Haar wavelets and multiresolution analysis | PDF |
| Hamiltonian flows, cotangent lifts, and momentum maps | PDF |
| Harmonic analysis on the \(p\)-adic numbers | PDF |
| Harmonic polynomials and the spherical Laplacian | PDF |
| Hensel's lemma, valuations, and \(p\)-adic numbers | PDF |
| Hermite functions | PDF |
| Infinite product measures | PDF |
| Integral operators | PDF |
| Jointly measurable and progressively measurable stochastic processes | PDF |
| Khinchin's inequality and Etemadi's inequality | PDF |
| Laguerre polynomials and Perron-Frobenius operators | PDF |
| Locally compact abelian groups | PDF |
| Lévy's inequality, Rademacher sums, and Kahane's inequality | PDF |
| Markov kernels, convolution semigroups, and projective families of probability measures | PDF |
| Martingales, Lévy's continuity theorem, and the martingale central limit theorem | PDF |
| Meager sets of periodic functions | PDF |
| Measure theory and Perron-Frobenius operators for continued fractions | PDF |
| Newton's identities and the pentagonal number theorem | PDF |
| Nonholomorphic Eisenstein series, the Kronecker limit formula, and the hyperbolic Laplacian | PDF |
| Norms of trigonometric polynomials | PDF |
| Notes on the KAM theorem | PDF |
| Orbital stability for NLS | PDF |
| Orthonormal bases for product measures | PDF |
| Oscillatory integrals | PDF |
| Polish spaces and Baire spaces | PDF |
| Positive definite functions, completely monotone functions, the Bernstein-Widder theorem, and Schoenberg's theorem | PDF |
| Projection-valued measures and spectral integrals | PDF |
| Proof of the pentagonal number theorem | PDF |
| Rademacher functions | PDF |
| Ramanujan's sum | PDF |
| Random trigonometric polynomials | PDF |
| Real reproducing kernel Hilbert spaces | PDF |
| Regulated functions and the regulated integral | PDF |
| Schwartz functions, Hermite functions, and the Hermite operator | PDF |
| Semicontinuous functions and convexity | PDF |
| Singular integral operators and the Riesz transform | PDF |
| Sobolev spaces in one dimension and absolutely continuous functions | PDF |
| Some fundamental results on modular forms | PDF |
| Spectral theory, Volterra integral operators and the Sturm-Liouville theorem | PDF |
| Stationary phase, Laplace's method, and the Fourier transform for Gaussian integrals | PDF |
| Subdifferentials of convex functions | PDF |
| Subgaussian random variables, Hoeffding's inequality, and Cramér's large deviation theorem | PDF |
| Tauber's theorem and Karamata's proof of the Hardy-Littlewood tauberian theorem | PDF |
| Test functions, distributions, and Sobolev's lemma | PDF |
| The \(C^\infty\) Urysohn lemma | PDF |
| The Banach algebra of functions of bounded variation and the pointwise Helly selection theorem | PDF |
| The Bernstein and Nikolsky inequalities for trigonometric polynomials | PDF |
| The Berry-Esseen theorem | PDF |
| The Bochner-Minlos theorem | PDF |
| The Dirac delta distribution and Green's functions | PDF |
| The Dunford-Pettis theorem | PDF |
| The Euler equations in fluid mechanics | PDF |
| The Fourier transform of holomorphic functions | PDF |
| The Fourier transform of spherical surface measure and radial functions | PDF |
| The Gelfand transform, positive linear functionals, and positive-definite functions | PDF |
| The Glivenko-Cantelli theorem | PDF |
| The Gottschalk-Hedlund theorem, cocycles, and small divisors | PDF |
| The Hamilton-Jacobi equation | PDF |
| The Heisenberg group and Hermite functions | PDF |
| The Hilbert transform on \(\mathbb{R}\) | PDF |
| The Kolmogorov continuity theorem, Hölder continuity, and the Kolmogorov-Chentsov theorem | PDF |
| The Kolmogorov extension theorem | PDF |
| The Legendre transform | PDF |
| The Lindeberg central limit theorem | PDF |
| The Poincaré-Dulac normal form theorem for formal vector fields | PDF |
| The Polya-Vinogradov inequality | PDF |
| The Pontryagin duals of \(\mathbb{Q}/\mathbb{Z}\) and \(\mathbb{Q}\) and the adeles | PDF |
| The Schwartz space and the Fourier transform | PDF |
| The Segal-Bargmann transform and the Segal-Bargmann space | PDF |
| The Stone-Čech compactification of Tychonoff spaces | PDF |
| The Voronoi summation formula | PDF |
| The Wiener algebra and Wiener's lemma | PDF |
| The Wiener-Pitt tauberian theorem | PDF |
| The cross-polytope, the ball, and the cube | PDF |
| The heat kernel on \(\mathbb{R}^n\) | PDF |
| The heat kernel on the torus | PDF |
| The inclusion map from the integers to the reals and universal properties of the floor and ceiling functions | PDF |
| The inhomogeneous heat equation on \(\mathbb{T}\) | PDF |
| The law of the iterated logarithm | PDF |
| The narrow topology on the set of Borel probability measures on a metrizable space | PDF |
| The nonlinear Schrödinger equation is Hamiltonian | PDF |
| The one-dimensional periodic Schrödinger equation | PDF |
| The profinite completion of the integers, the \(p\)-adic integers, and Prüfer \(p\)-groups | PDF |
| The singular value decomposition of compact operators on Hilbert spaces | PDF |
| The spectra of the unilateral shift and its adjoint | PDF |
| The spectrum of a self-adjoint operator is a compact subset of \(\mathbb{R}\) | PDF |
| The symmetric difference metric | PDF |
| The theorem of F. and M. Riesz | PDF |
| The uniform metric on product spaces | PDF |
| The weak and strong laws of large numbers | PDF |
| The weak topology of locally convex spaces and the weak-* topology of their duals | PDF |
| Topological spaces and neighborhood filters | PDF |
| Total variation, absolute continuity, and the Borel \(\sigma\)-algebra of \(C(I)\) | PDF |
| Trace class operators and Hilbert-Schmidt operators | PDF |
| Unbounded operators in a Hilbert space and the Trotter product formula | PDF |
| Uncertainty principles and compressed sensing | PDF |
| Unordered sums in Hilbert spaces | PDF |
| Vinogradov's estimate for exponential sums over primes | PDF |
| Vitali coverings on the real line | PDF |
| Watson's lemma and Laplace's method | PDF |
| Weak symplectic forms and differential calculus in Banach spaces | PDF |
| What is a Costas array? | PDF |
| What is the domain of a solution of an ODE? | PDF |
| Wiener measure and Donsker's theorem | PDF |
| Zygmund's Fourier restriction theorem and Bernstein's inequality | PDF |