Math 641-600 Midterm Review

The in-class part of the midterm will consist of the following: statements of theorems and definitions; short problems or propositions similar to homework problems or examples done in class; and either a critical part or sketch of a proof for one of the major theorems proved.

The take-home test will have longer computations, proofs, or problems. It is due on Monday, October 19, at 4 pm. You may bring the test to me directly (Blocker 611D) or put it in my mailbox on the second floor.

Linear algebra

Inner products & norms

• Subspaces, orthogonal complements
• Orthogonal sets of vectors, the Gram-Schmidt procedure
• Least squares, minimization problems, projections, normal equations

Self-adjoint matrices & their properties

• Spectral theorem
• Estimation of eigenvalues
  • Maximum principle
  • The Courant-Fischer theorem
• The Fredholm Alternative

Function spaces

Banach spaces and Hilbert spaces

• Convergent sequence, Cauchy sequence, complete spaces - Hilbert spaces and Banach spaces
• Special (complete) spaces - l^p, L^p (1 ≤ p ≤ ∞), C[a,b], C^k[a,b], Sobolev space Hⁿ[a,b]
• Continuous functions on [0,1].

Lebesgue Integration

• Lebesgue integral, sets of measure 0 and L^p spaces
• Density of continuous functions in L^p[a,b], 1 ≤ p < ∞
• Monotone convergence theorem and dominated convergence theorem (skip Fubini's theorem)

Orthonormal sets and expansions

• Minimization problems, least squares, normal equations
• Complete sets of orthogonal/orthonormal functions, Parseval's identity, other conditions equivalent to completeness of a set
• Dense sets and completeness
• Completeness of polynomials in L², orthogonal polynomials

Approximation of continuous functions

• Modulus of continuity, linear spline approximation
• Bernstein polynomials
• Weierstrass Approximation Theorem

Fourier series

• Riemann-Lebesgue Lemma
• Sketch proof of pointwise convergence