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), 2.2.1-2.2.3. It will also cover
the notes on the
pointwise convergence of Fourier Series. The test will be
composed of three 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 red below. The third part will be
take-home. The problems it will have will be longer computations or
proofs and might involve simple numerical problems 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 (discrete and continuous), minimization problems,
projections, normal equations
- Spectral Decomposition Theorem, invariant subspaces
- The Fredholm Alternative
- The Courant (or Courant-Fischer) Minimax
Theorem
- Singular value decomposition (SVD)
Hilbert & Banach 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]
- Completeness of C[a,b], uniform convergence
- Complete sets of orthogonal functions, Parseval's identity
- Orthogonal polynomials & weight functions
- The Weierstrass Approximation
Theorem, Bernstein polynomials, modulus of continuity
- Fourier series, the Riemann-Lesbegue Lemma, and the Fourier Convergence Theorem
Updated 10/12/06 (fjn).