SYLLABUS FOR THE PRELIMINARY EXAMINATION IN APPLIED MATHEMATICS
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1. Applications of elementary Banach space theory:
   Elementary banach space theory, including conjugate spaces,
   Hahn-Banach Theorem; weak topologies, Banach-Alaoglu Theorem,
   reflexivity; bounded linear operators, Neumann Theorem on inverses,
   spectrum, theorem on the Fredholm alternative; Banach's contraction
   mapping theorem and its applications, especially in differential
   equations (the initial value problem and two-point boundary-value
   problems) and integral equations.
2. Function Spaces:
   Distributions, weak derivatives, Sobolev spaces; Green's functions.
3. Hilbert space:
   Hilbert space theory up to the spectral theorem for compact Hermitian
   operators; Raileigh-Ritz method for computing eigenvalues;
   orthonormal expansions, particularly the classical Fourier series;
   applications to the Sturm-Liouville problem; Fourier transform.
4. Differential calculus in normed linear spaces:
   Differential calculus, including Gateaux and Frechet derivatives,
   mean value theorem, partial derivatives, the chain rule;
   implicit function theorems, inverse function theorems; extremal problems,
   Lagrange multipliers; applications - the method of steepest descent,
   Newton's method (including the Kantorovich theorem).
5. Approximation and computational methods:
   Discretization, linearization, Galerkin method, Ritz method, method
   of steepest descent, variational methods, the method of collocation,
   iterative methods, methods based on the Neumann series, Newton's method;
   finite difference methods and numerical integration; numerical linear
   algebra and the solution of ordinary and partial differential equations.