Skip to content

Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Spring 2020 • Question 6

Question

Math 6210 • Real Analysis Spring 2020 • Question 6

Let \(\mathcal H\) be a Hilbert space and \(P:\mathcal H\to \mathcal H\) a self-adjoint operator, i.e. \(\langle Px,y\rangle =\langle x,Py\rangle \) for all \(x,y\in \mathcal H\). Also assume that \(P^2=P\). Prove that \(P\) is orthogonal projection to a closed subspace. Note that we are not assuming that \(P\) is bounded.

Raw LaTeX

Let \(\mathcal H\) be a Hilbert space and \(P:\mathcal H\to\mathcal H\)
a self-adjoint operator, i.e. \(\langle Px,y\rangle=\langle x,Py\rangle\) for
all \(x,y\in\mathcal H\). Also assume that \(P^2=P\). Prove that \(P\) is
orthogonal projection to a closed subspace. Note that we are not assuming
that \(P\) is bounded.

\newpage

Topic Tags

Hilbert Spaces and Orthogonality