## Math 641-600 Midterm Review — Fall 2017

The midterm will consist of an in-class part, which will be given on Friday, Oct. 13, and a take-home part. It will cover sections 1.1-1.3, 2.1, 2.2.1-2.2.3. It will also cover the material done in class and covered in my notes, starting with "Adjoints,...(Courant Fischer)" and ending with "Pointwise convergence of Fourier series." See Class notes on my web page.

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 Friday, October 20.

### Linear algebra

Inner products & norms
• Subspaces, orthogonal complements
• Orthogonal sets of vectors, the Gram-Schmidt procedure
• Least squares, minimization problems, projections, normal equations
• Spectral theorem
• Estimation of eigenvalues
• Maximum principle
• The Courant-Fischer theorem; be able to sketch a proof.
• The Fredholm Alternative

### Function spaces

Banach spaces and Hilbert spaces
• Convergent sequence, Cauchy sequence, 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 $H^1[a,b]$.
Lebesgue Integration
• Lebesgue integral, sets of measure 0 and Lp spaces
• Density of continuous functions in Lp[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
• Riemann-Lebesgue Lemma; be able to prove.
• Sketch proof of pointwise convergence
• 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.
Updated 10/10/2017 (fjn).