## 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 andvector 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.