Math 641-600 Midterm Review
The midterm will be given on October 18 and will consist of an in-class part and a take-home part. The test will cover sections 1.1-1.5 (but not 1.5.1), 2.1 (all), 2.2.1-2.2.2. It will also cover the material done in class on the Lebesgue integral.
The in-class test will consist of the following: statements of theorems and definitions; short problems or propositions similar to homework problems or examples done in class; and a proof of one of the major theorems highlighted in red below. The take-home test will have longer computations or proofs and problems involving requiring a computer.
Linear algebra
- Inner product spaces & normed linear spaces
- Schwarz's inequality
- Trangle inequaity
- Subspaces, orthogonal complements
- Orthogonal sets of vectors, the Gram-Schmidt procedure
- Least squares, minimization problems, projections, normal equations
- Spectral Decomposition Theorem, invariant subspaces
- The maximum principle for eigenvalues
- The Fredholm Alternative
- The Courant (or Courant-Fischer) Minimax Theorem
- Singular value decomposition (SVD)
Banach & Hilbert spaces
- Convergent sequence, Cauchy sequance, complete spaces - Hilbert spaces and Banach spaces
- Lebesgue integrals, sets of measure 0, measurable functions, the Dominated Convergence Theorem
- Special spaces - Lp (1 ≤ p ≤ ∞), C[a,b],
Ck[a,b], L2, Sobolev space Hn[a,b]
- The Completeness of C[a,b], uniform convergence, modulus of continuity
- Complete sets of orthogonal functions, Parseval's identity
- Orthogonal polynomials & weight functions
- The Weierstrass Approximation Theorem, Bernstein polynomials,
- Completeness orthogonal polynomials
Updated 10/14/07 (fjn).