Math 641-600 — Fall 2014

• Read sections 3.5, 3.6, and 4.1.
• Do the following problems.
  1. Section 3.4: 2(c,d), 3
  2. Section 3.5: 1(b), 2(a)
  3. Section 3.6: 1(a), 5

  4. 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.)
     1. Show that M and M^⊥ are both invariant under K.
     2. Show that K restricted to M^⊥ is compact.
     3. Show that either M^⊥ = {0} or that it is the eigenspace for λ = 0.
     4. Show that one may choose a complete, orthonormal set from among the eigenvectors of K.

  5. 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 $B_r:=\{u\in C[0,1]\,|\, \|u\|\le r\}$.
     1. Show that $F: B_1\to B_{1/2}\subset B_1$.
     2. Show that $F$ is Lipschitz continuous on $B_1$, with Lipschitz constant $0<\alpha<1$ -- i.e., $\|F[u]-F[v]\|\le \alpha \|u-v\|$.
     3. Show that $F$ has a fixed point in $B_1$.

Updated 11/13/2014.