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.
[4] P. Lax, Functional Analysis.
[5] A.N. Kolmogorov and S.V. Fomin, Introductory Real Analysis.
[6] H. Royden, Real Analysis.

• 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 Arzelà-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.