Math 641-600 Midterm Review — Fall 2017
The midterm will consist of an in-class part, which will be given on
Friday, Oct. 13, and a take-home part. It will cover sections 1.1-1.3,
2.1, 2.2.1-2.2.3. It will also cover the material done in class and
covered in my notes, starting with "Adjoints,...(Courant Fischer)" and
ending with "Pointwise convergence of Fourier series." 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
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 Friday, October 20.
products & norms
- Subspaces, orthogonal complements
- Orthogonal sets of vectors, the Gram-Schmidt procedure
- Least squares, minimization problems, projections, normal
Self-adjoint matrices & their properties
- Spectral theorem
- Estimation of eigenvalues
- Maximum principle
- The Courant-Fischer theorem; be able to sketch a proof.
- The Fredholm Alternative
Updated 10/10/2017 (fjn).
spaces and Hilbert spaces
- Convergent sequence, Cauchy sequence, complete spaces - Hilbert
spaces and Banach spaces
- Special (complete) spaces — $\ell^p, L^p\ (1 \le p \le
\infty),\ C[a,b], C^k[a,b]$, and Sobolev space $ H^1[a,b]$.
- 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
(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; be able to establish completeness for specific sets of
Approximation of continuous functions
- Modulus of continuity, linear spline approximation; be able prove
that $\|f- s_f\|\le \omega(f,\delta)$.
- Bernstein polynomials; be able to define these and to show that
they span the appropriate space of polynomials.
- Weierstrass Approximation Theorem; be able to sketch a proof,
given necessary properties of the Bernstein polynomials.
- Riemann-Lebesgue Lemma; be able to prove.
- Sketch proof of pointwise convergence
- Be able to find the Fourier series for a given function and to use
Parseval's equation to sum series or estimate $L^2$ errors.