Math 641-600 Midterm Review — Fall 2019

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 at 4 pm on Wednesday, October 16, 2019.

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; be able to sketch a proof.

Function spaces

Banach spaces and Hilbert spaces

• Convergent sequences, Cauchy sequences, complete spaces - Hilbert spaces and Banach spaces.
• Special (complete) spaces — $\ell^p, L^p\ (1 \le p \le \infty),\ C[a,b], C^k[a,b]$, and Sobolev space $ H^1[a,b]$. Be able to prove completeness for the spaces given in the notes or in homework assignments.

Lebesgue Integration

• Lebesgue measure, 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; be able to establish completeness for specific sets of orthogonal polynomials.

Approximation of continuous functions

• Modulus of continuity, linear spline approximation; be able prove that $\|f- s_f\|\le \omega(f,\delta)$.
• Bernstein polynomials. Be able to define these and to show that they span the appropriate space of polynomials.
• Weierstrass Approximation Theorem. Be able to sketch a proof, given necessary properties of the Bernstein polynomials.

Fourier series

• Be able to find the Fourier series for a given function and to use Parseval's equation to sum series or estimate $L^2$ errors.
• Riemann-Lebesgue Lemma. Be able to prove it.
• Pointwise convergence of Fourier series. Be able to sketch a proof of it.
• Completeness of $\{e^{inx}\}_{n=-\infty}^\infty$ in $L^2[-\pi,\pi]$. Be able to sketch a proof of it.