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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6420 • PDE

Fall 2016 • Question 1

Question

Math 6420 • PDE Fall 2016 • Question 1

Consider the operator \(T:\ell ^2\to \ell ^2\) defined for \(x=(\xi _j)\in \ell ^2\) by \[ Tx=(0,\xi _1\alpha _1,\xi _2\alpha _2,\ldots ), \] for some sequence \((\alpha _j)\) with \(\alpha _j\to 0\). Show that \(T\) is compact.

Raw LaTeX

Consider the operator \(T:\ell^2\to\ell^2\) defined for \(x=(\xi_j)\in\ell^2\) by
\[
  Tx=(0,\xi_1\alpha_1,\xi_2\alpha_2,\ldots),
\]
for some sequence \((\alpha_j)\) with \(\alpha_j\to0\). Show that \(T\) is compact.

Topic Tags

Linear Programming and Optimization Sequence Spaces