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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6420 • PDE

Fall 2006 • Question 1

Question

Math 6420 • PDE Fall 2006 • Question 1

  1. State the Riesz Representation Theorem (a proof is not required).
  2. Use the Riesz Representation theorem to prove that a bounded linear operator on a Hilbert Space has an adjoint. Prove that the adjoint is a bounded linear operator on the Hilbert space.
  3. Find the adjoint for the operator \[ (Ku)(y)=\int _0^1 \min (x,y)u(x)\,dx. \]

Raw LaTeX

\begin{enumerate}
\item[(a)] State the Riesz Representation Theorem (a proof is not required).
\item[(b)] Use the Riesz Representation theorem to prove that a bounded linear operator on a
Hilbert Space has an adjoint. Prove that the adjoint is a bounded linear operator on the Hilbert
space.
\item[(c)] Find the adjoint for the operator
\[
  (Ku)(y)=\int_0^1 \min(x,y)u(x)\,dx.
\]
\end{enumerate}

Topic Tags

Adjoint Operators and Riesz Representation Bounded Linear Operators Dual Spaces and Bounded Linear Functionals Linear Programming and Optimization