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
-
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 - $\ell^p$, $L^p$, $C[a,b]$,
$C^k[a,b]$, 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).