Math 641-600 — Fall 2013

1. Finish the proof of the Projection Theorem: If for every $f\in \mathcal H$ there is a $p\in V$ such that $\|p-f\|=\min_{v\in V}\|v-f\|$ then $V$ is closed.
2. Prove this: If $L:\mathcal H\to \mathcal H$ is a bounded linear transformation, then $\overline{R(L)} = N(L^*)^\perp$.
3. Let $\mathcal H$ be a Hilbert space of functions that are defined on $[0,1]$. In addition, suppose that $\mathcal H \subset C[0,1]$, with $\|f\|_{C[0,1]} \le K\|f\|_{\mathcal H}$ for all $f\in \mathcal H$. (The Sobolev space $H^1$ has this property.)
   1. Show that the point-evaluation functional $\Phi_x(f) =f(x)$ is a bounded linear functional on $\mathcal H$.
   2. Let $x$ be fixed. Show that there is a kernel $k(x,y)\in \mathcal H$ such that \[ \Phi_x(f)=f(x) = \langle f,k(x,\cdot)\rangle \] (The kernel $k(x,y)$ is called a reproducing kernel and $\mathcal H$ is called a reproducing kernel Hilbert space.)
   3. For $x,z$ fixed, show that $k(z,x) = \langle k(z,\cdot),k(x,\cdot)\rangle$. In addition, let $\{x_j\}_{j=1}^n$ be any finite set of distinct points (i.e. $x_j\ne x_k$ if $j\ne k$) in $[0,1]$, show that the matrix $G_{jk} = k(x_k,x_j)$ is positive semidefinite; that is, $\sum_{j,k}c_k\overline{c_j}k(x_k,x_j)\ge 0$
   4. Suppose the matrix $G$ is positive definite and therefore invertible. Let $f\in \mathcal H$. Show that there are unique coefficients $\{c_j\}_{j=1}^n$ such that $s(x) =\sum_{j=1}^n k(x_j,x)c_j$ interpolates $f$ at the $x_j$'s.
4. Consider the finite rank (degenerate) kernel k(x,y) = φ₁(x)ψ₁(y) + φ₂(x)ψ₂(y), where φ₁ = 2x-1, φ₂ = x², ψ₁ = 1, ψ₂ = x. Let Ku= ∫₀¹ k(x,y)u(y)dy. Assume that L = I-λ K has closed range,

   Alermative set of functions: Keep φ₁, φ₂, and ψ₁ the same. For ψ₂, use ψ₂ = 4x − 3.

   1. For what values of λ does the integral equation
      u(x) - λ∫₀¹ k(x,y)u(y)dy =f(x)
      have a solution for all f ∈ L²[0,1]?
   2. For these values, find the solution u = (I − λK)⁻¹f — i.e., find the resolvent.
   3. For the values of λ for which the equation does not have a solution for all f, find a condition on f that guarantees a solution exists. Will the solution be unique?

5. Consider the Hilbert space $\ell^{\,2}$. Let $S=\{\{a_j\}_{j=1}^\infty \in \ell^{\,2}\colon \sum_{j=1}^\infty (1+j^2)\,|a_j|^2\le 1 \}$. Show that $S$ is a compact subset of $\ell^{\,2}$.

Updated 11/14/2013.