## Syllabus for the Ph.D. 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

1. Jerrold E. Marsden and Michael J. Hoffman, Elementary Classical Analysis, Freeman 1993.
2. W. Rudin, Principles of Mathematical Analysis, 3rd edition, McGraw-Hill, 1976.
3. Michael Spivak, Calculus on Manifolds, Addison-Wesley 1965.
4. T. Apostol, Mathematical Analysis, Addison-Wesley 1974.
The Analysis syllabus is downloadable as a pdf file.

## Syllabus for the Ph.D. Preliminary Examination in Linear Algebra

Topics

Vector Spaces: subspaces, linear independence, bases, dimension, isomorphism, linear functionals, dual space, bilinear forms.

Matricies 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.

Inner Product Spaces: inner products, norms, orthogonality, projections, orthogonal complement, orthonormal basis, Gram-Schmidt orthogonalization, linear functionals, isometries, normal operators, spectral theory.

REFERENCES

1. P. R. Halmos, Finite Dimensional Vector Spaces, Springer, 1993
2. R. Horn and C. Johnson, Matrix Analysis, Cambridge, 1999
3. Peter Lax, Linear Algebra, Wiley-Interscience, 1997.
4. K. Hoffman and R. Kunze, Linear Algebra, 2nd edition, Prentice Hall, 1972.
The Linear Algebra Syllabus is downloadable as a pdf file.