Math 641600 Midterm Review, Fall 2016
The midterm will be given on Thursday, Oct. 13. It will cover sections
1.11.4, 2.1, 2.2.12.2.2. 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 GramSchmidt procedure
 Least squares, minimization problems, projections, normal
equations

Selfadjoint matrices & their properties
 Spectral theorem
 Estimation of eigenvalues
 Maximum principle
 The CourantFischer 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 L^{p} spaces
 Density of continuous functions in L^{p}[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).