Math 641600 Midterm Review — Fall 2017
The midterm will consist of an inclass part, which will be given on
Friday, Oct. 13, and a takehome part. It will cover sections 1.11.3,
2.1, 2.2.12.2.3. 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 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 Friday, October 20.
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; be able to sketch a proof.
 The Fredholm Alternative
Function spaces
 Banach
spaces and Hilbert spaces
 Convergent sequence, Cauchy sequence, complete spaces  Hilbert
spaces and Banach spaces
 Special (complete) spaces — $\ell^p, L^p\ (1 \le p \le
\infty),\ C[a,b], C^k[a,b]$, and 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
(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; be able to establish completeness for specific sets of
orthogonal polynomials.

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.

Fourier series
 RiemannLebesgue 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.
Updated 10/10/2017 (fjn).