Math 641-600 Midterm Review
The midterm will be given on Wednesday, Oct. 15 and will consist of an
in-class part and a take-home part. 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, starting with "Adjoints,..." and ending with
Approximation of continuous functions." 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 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 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 - lp, Lp (1 ≤
p ≤ ∞), C[a,b], Ck[a,b], Sobolev space
Hn[a,b]
- Continuous functions on [0,1].
- Lebesgue
Integration
- Lebesgue measure, measurable functions, Lebesgue sums and
Lebesgue integral
- Density of continuous functions in Lp[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 L2, orthogonal polynomials
- Approximation
of continuous functions
- Modulus of continuity, linear spline approximation
- Bernstein polynomials
- Weierstrass Approximation Theorem
Updated 10/10/2014 (fjn).