## Math 641-600 Midterm Review

The midterm will consist of an in-class part, which will be given on Wednesday, Oct. 14, and a take-home part. It will cover sections 1.1-1.4, 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,..." 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 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
• 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 - lp, Lp (1 ≤ p ≤ ∞), C[a,b], Ck[a,b], Sobolev space Hn[a,b]
• Continuous functions on [0,1].
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
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
Updated 10/9/2015 (fjn).