## Math 641-600 (Fall 2014) — Final Exam Review

The final will be given from 8 am to 10 am on Tuesday, December 16, 2014, in our usual classroom. The test will cover sections 2.2.3, 2.2.7, 3.2-3.6, 4.1, 4.2 4.3.1-4.3.2 (2nd order operators, only), 4.5 (pp. 161-163) in the text, class notes, and any material covered in class. The test will be composed of two parts. The first part will consist of statements of theorems and definitions; the second will have short problems or propositions similar to homework problems or examples done in class, as well as a proof of one of the major theorems highlighted in blue below.

### Approximation tools

• Fourier series, point-wise convergence, Parseval's theorem (skip DFT)
• Finite elements, spline spaces Sh(k,r), linear splines, cubic interpolating splines, Galerkin method (skip B-splines)

### Operators

• Bounded operators
• B(H), bounded operators on H, continuous linear functionals
• Examples: finite-rank operators, Hilbert-Schmidt operators, norms of Hilbert-Schmidt operators
• Closed subspaces: null spaces, orthogonal complements, etc.
• Projection Theorem
• The Riesz Representation Theorem
• Weak form of a boundary value problem
• Fredholm Alternative, applications to solving integral equations
• Compact operators
• C(H) is closed in B(H).
• Finite rank operators, and Hilbert-Schmidt operators,
• Closed range theorem, Fredholm alternative for L = I − λK, application to integral equations
• Spectral theory for compact operators
• Eigenvalues, eigenspaces
• Completeness of eigenfunctions
• Resolvents and resolvent kernels, solutions via eigenfunction expansions
• Contraction Mapping Theorem, Neumann series

### Distributions and Differential Operators

• D, convergence in D, "bump" function, D′, convergence in D′, δ function, derivatives of distributions
• Green's functions