Math 641600 Midterm Review
The midterm will be given on Wednesday, Oct. 15 and will consist of an
inclass part and a takehome part. It will cover sections 1.11.4,
2.1, 2.2.1, 2.2.2. It will also cover the material done in class and
covered in my notes, starting with "Adjoints,..." and ending with
Approximation of continuous functions." See Class notes on
my web page.
The inclass 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 takehome test will have longer computations, proofs, or
problems. It is due on Tuesday, October 21, 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 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  l^{p}, L^{p} (1 ≤
p ≤ ∞), C[a,b], C^{k}[a,b], Sobolev space
H^{n}[a,b]
 Continuous functions on [0,1].
 Lebesgue
Integration
 Lebesgue measure, measurable functions, Lebesgue sums and
Lebesgue integral
 Density of continuous functions in L^{p}[a,b], 1 ≤ p <
∞
 Monotone and dominated convergence theorems (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 L^{2}, orthogonal polynomials
 Approximation
of continuous functions
 Modulus of continuity, linear spline approximation
 Bernstein polynomials
 Weierstrass Approximation Theorem
Updated 10/10/2014 (fjn).