Skip to content

Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6420 • PDE

Spring 2020 • Question 1

Question

Math 6420 • PDE Spring 2020 • Question 1

Consider the sequence of linear operators \((U_n)\) where \(U_n:H\to H\), \(H\) is a Hilbert space and the \(U_n\) are unitary, i.e. \(U_n^*U_n=I\), where \(I\) is the identity operator on \(H\). Show that if \(U_n\to U\) in the operator norm, the limit \(U\) must also be unitary, i.e. \(U^*U=I\).

Raw LaTeX

Consider the sequence of linear operators \((U_n)\) where \(U_n:H\to H\), \(H\) is a Hilbert
space and the \(U_n\) are unitary, i.e. \(U_n^*U_n=I\), where \(I\) is the identity operator on
\(H\). Show that if \(U_n\to U\) in the operator norm, the limit \(U\) must also be unitary,
i.e. \(U^*U=I\).

Topic Tags

Bounded Linear Operators Linear Programming and Optimization Matrix Norms and Induced Norms Orthogonal and Unitary Matrices