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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6420 • PDE

Fall 2004 • Question 4

Question

Math 6420 • PDE Fall 2004 • Question 4

  1. Prove that the eigenvalues of a self-adjoint linear operator acting in a Hilbert space \(H\) are real and that the eigenfunctions of distinct eigenvalues are orthogonal.
  2. Prove that if \(L\) is a bounded linear operator in \(H\) then the adjoint \(L^*\) is also a bounded linear operator.

Raw LaTeX

\begin{enumerate}
\item[(a)] Prove that the eigenvalues of a self-adjoint linear operator acting in a Hilbert
space \(H\) are real and that the eigenfunctions of distinct eigenvalues are orthogonal.
\item[(b)] Prove that if \(L\) is a bounded linear operator in \(H\) then the adjoint \(L^*\) is
also a bounded linear operator.
\end{enumerate}

Topic Tags

Adjoint Operators and Riesz Representation Bounded Linear Operators Eigenvalues, Spectral Radius, and Gershgorin Theorem Linear Programming and Optimization