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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6420 • PDE

Fall 2008 • Question 3

Question

Math 6420 • PDE Fall 2008 • Question 3

Prove that if a linear operator \(K\) can be approximated (in the operator norm) by a sequence of compact operators \(K_n\), with, \[ \lim _{n\to \infty }\lVert K_n-K\rVert =0, \] then \(K\) is a compact operator.

Raw LaTeX

Prove that if a linear operator \(K\) can be approximated (in the operator norm) by a sequence
of compact operators \(K_n\), with,
\[
  \lim_{n\to\infty}\lVert K_n-K\rVert=0,
\]
then \(K\) is a compact operator.

Topic Tags

Bounded Linear Operators Compact Operators Linear Programming and Optimization Matrix Norms and Induced Norms