# Graduate Handbook

### Syllabus for the PhD Preliminary Examination in Analysis

#### Topics

*Metric spaces:* open and closed sets, convergence, compactness, connectedness, completeness, continuity, uniform continuity, uniform convergence, equicontinuity and the Ascoli-Arzela Theorem, contraction mapping theorem

*Single variable analysis:* numerical sequences and series, differentiation, mean value theorem, Taylor's theorem, function series and power series, uniform convergence and differentiability, Weierstrass approximation theorem, Riemann integral, sets of measure zero

*Several variables analysis:* differentiability, partial derivatives, inverse and implicit function theorems, iterated integrals, Jacobians, change of variable in multiple integrals

*Vector analysis:* Stokes theorem, Green's theorem, divergence theorem

#### REFERENCES

- Jerrold E. Marsden and Michael J. Hoffman,
*Elementary Classical Analysis*, Freeman, 1993 - W. Rudin,
*Principles of Mathematical Analysis*, 3rd edition, McGraw-Hill, 1976 - Michael Spivak,
*Calculus on Manifolds*, Addison-Wesley, 1965 - T. Apostol,
*Mathematical Analysis*, Addison-Wesley, 1974

The analysis syllabus is downloadable as a pdf file.

### Syllabus for the PhD Preliminary Examination in Linear Algebra

#### Topics

*Vector spaces*: subspaces, linear independence, bases, dimension,

isomorphism, linear functionals, dual space, adjoints, inverses and reducibility*Matrices and linear transformations*: range, kernel, determinants,

isomorphisms, change of basis, eigenvalues, eigenvectors, minimax

Theory of eigenvalues, Gersgorin discs, minimal polynomial,

Cayley-Hamilton theorem, similarity, polar and singular value

decomposition, spectral theorem, Jordan cannonical forms.

Hermitian, symmetric, and positive definite matricies. Matrix and

vector norms, convergence of sequences, powers etc. of matrices*Inner product spaces*: inner products, norms, orthogonality,

projections, orthogonal complement, orthonormal basis, Gram-Schmidt

orthogonalization, linear functionals, isometries, normal

operators, spectral theory, basic inequalities such as Cauchy Schwarz

#### REFERENCES

- P.R. Halmos, Finite Dimensional Vector Spaces, Springer, 1993
- R. Horn and C. Johnson, Matrix Analysis, Cambridge, 1999
- Peter Lax, Linear Algebra, Wiley-Interscience, 1997.
- K. Hoffman and R. Kunze, Linear Algebra, 2nd ed., Prentice

Hall, 1972 - P. N. de Souza and J.-G. Silva, Berkeley Problems in Mathematics, Springer,

Berlin, 2004, (Chapter 7: Linear Algebra)

The linear algebra syllabus is downloadable as a pdf file.

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