University of Pittsburgh

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

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

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

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 ed., Prentice
Hall, 1972
5. P. N. de Souza and J.-G. Silva, Berkeley Problems in Mathematics, Springer,
Berlin, 2004, (Chapter 7: Linear Algebra)

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