# Math 641-600 — Fall 2014

## Current Assignment

**Assignment 10** - Due Wednesday, December 3.
- Read sections 3.5, 3.6, and 4.1.
- Do the following problems.

- Section 3.4: 2(c,d), 3
- Section 3.5: 1(b), 2(a)
- Section 3.6: 1(a), 5

- Let K be a compact, self-adjoint operator and let M be the span
of the set of eigenvectors {φ
_{j}} corresponding to all
eigenvalues λ_{j} ≠ 0. (Note: both M and
M^{⊥} may be infinite dimensional.)
- Show that M and M
^{⊥} are both invariant under K.
- Show that K restricted to M
^{⊥} is compact.
- Show that either M
^{⊥} = {0} or that it is the
eigenspace for λ = 0.
- Show that one may choose a complete, orthonormal set from among
the eigenvectors of K.

- Let $F:C[0,1]\to C[0,1]$ be defined by $F[u](t) :=
\int_0^1(2+st+u(s)^2)^{-1}ds$, $0\le t\le 1$. Let $\| \cdot
\|:=\|\cdot \|_{C[0,1]}$. Let $B_r:=\{u\in C[0,1]\,|\, \|u\|\le
r\}$.
- Show that $F: B_1\to B_{1/2}\subset B_1$.
- Show that $F$ is Lipschitz continuous on $B_1$, with Lipschitz
constant $0<\alpha<1$ -- i.e., $\|F[u]-F[v]\|\le \alpha \|u-v\|$.
- Show that $F$ has a fixed point in $B_1$.

Updated 11/13/2014.