## Math 641-600 Midterm Review, Fall 2016

The midterm will be given on Thursday, Oct. 13. It will cover sections 1.1-1.4, 2.1, 2.2.1-2.2.2. It will also cover the material done in class and covered in my notes. (See the class notes on my web page.)

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 proof, or a sketch of a proof, for one of the major theorems covered in class.

### 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 - $\ell^p$, $L^p$, $C[a,b]$, $C^k[a,b]$, 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
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 $L^2$, orthogonal polynomials
Approximation of continuous functions
• Modulus of continuity, linear spline approximation
• Bernstein polynomials
• Weierstrass Approximation Theorem
Updated 10/9/2016 (fjn).