The principal reference for this syllabus is:

[1] J.K. Hunter and B. Nachtergaele, Applied Analysis.

With the exception of the topics from advanced calculus, a question may
appear on the exam if and only if the topic is covered in one of the sections
of [1] listed below. Recommended supplemental references include:

[2] A. Friedman, Foundations of Modern Analysis.
[3] W. Fulks, Advanced Calculus.
[4] P. Lax, Functional Analysis.
[5] A.N. Kolmogorov and S.V. Fomin, Introductory Real Analysis.
[6] H. Royden, Real Analysis.

Advanced Calculus

  • Integration of functions of several variables: line and volume integrals in 2D, line, surface, and volume integrals in 3D.
  • Differentiation: gradient, curl, divergence, Jacobian. Connection between rotation-free vector fields and potential fields.
  • Partial integration, Green’s theorems, Stokes’ theorem, Gauss’ theorem. The consequences of these theorems for vector fields that are divergence or rotation free.
  • The concepts max, min, sup, inf, lim sup, lim inf, lim.
  • Convergence criteria for sequences and series.

Metric and Normed Spaces:

  • The topology of metric spaces. Normed spaces. Cauchy sequences. Compactness. Completeness. Sections 1.1 – 1.7.
  • Basic properties of the space of continuous functions on a metric space. The ArzelĂ -Ascoli theorem. Sections 2.1, 2.2, and 2.4.
  • The contraction mapping theorem. Sections 3.1 and 3.3.

Banach Spaces:

  • Bounded linear maps between Banach spaces, different topologies on the set of bounded operators between Banach spaces. The kernel and the range of a linear map. Connection between an operator being coercive and having closed range. The exponential of a bounded operator on a Banach space. Sections 5.1 – 5.5.
  • The dual of a Banach space. The Hahn-Banach theorem. Compactness of the unit ball in the weak topology on reflexive spaces. Section 5.6 (weak-* convergence and the full Banach-Alaoglu theorems are not included).

Hilbert Spaces (separable spaces only):

  • Orthogonal sets and orthonormal bases. Bessel’s inequality. Parallelogram law and polarization identity. Sections 6.1 – 6.3.
  • Riesz representation theorem. Section 8.2.
  • Fourier series. Parseval identity. Convolutions. Section 7.1.
  • Sobolov spaces on the \(d\)-dimensional torus. Sobolev embedding. Section 7.2.

Linear Operators on Hilbert Spaces:

  • The adjoint of an operator. Self-adjoint, normal, and unitary operators. Projections. \(\overline{\text{ran} A} = (\ker A^*)^\perp\). Fredholm alternative, in particular for the case of the identity plus a compact operator. Sections 8.1, 8.3, 8.4.
  • The spectrum of general operators on a Hilbert space. Basic properties of the resolvent operator. The spectral theorem for self-adjoint compact operators. Functions of operators. Sections 9.2, 9.3, 9.4, 9.5.

Measure Theory, Integration, and Lp-spaces:

  • Basic properties of Lebesgue measure. Nullsets. “Almost everywhere” and “essential supremum”. Sections 12.1 and 12.2 (but only the concepts listed here).
  • Definition of the Lebesgue integral. Section 12.3.
  • Convergence theorems: Fatou’s lemma, Monotone convergence, Lebesgue dominated convergence. Section 12.4.
  • Fubini’s theorem (with respect to Lebesgue measure on \(\mathbb{R}^d\) only). Section 12.5.
  • Definition of \(L^p(X,\mu)\) for a measure space \((X,\mu)\). Hölder’s and Minkowski’s inequalities. The dual of \(L^p(X,\mu)\) for \(p\in[1,\infty)\). Density of simple functions, and compactly supported smooth functions in \(L^p(\mathbb{R}^d)\). Sections 12.6, 12.8, and parts of 12.7.