Skip to content

Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Spring 2008 • Question 3

Question

Math 6210 • Real Analysis Spring 2008 • Question 3

Let \(H\) be a Hilbert space, and let \(T:H\to H\) be a self adjoint linear operator, with finite dimensional range \(V=T(H)\). Prove that \(T\) is a compact operator, and that \(V=K^\perp \), where \(K\) is the kernel of \(T\).

Raw LaTeX

Let \(H\) be a Hilbert space, and let \(T:H\to H\) be a self adjoint
linear operator, with finite dimensional range \(V=T(H)\). Prove that \(T\) is a
compact operator, and that \(V=K^\perp\), where \(K\) is the kernel of \(T\).

Topic Tags

Bounded Linear Operators Compactness in Metric and Normed Spaces Hilbert Spaces and Orthogonality