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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6420 • PDE

Spring 2016 • Question 2

Question

Math 6420 • PDE Spring 2016 • Question 2

Let \(X\) and \(Y\) be normed vector spaces. Show that a linear operator \(T:X\to Y\) is compact if and only if for every sequence \((x_n)\) with \(\lVert x_n\rVert \le 1\), the sequence \((Tx_n)\) has a convergent subsequence.

Raw LaTeX

Let \(X\) and \(Y\) be normed vector spaces. Show that a linear operator \(T:X\to Y\) is compact
if and only if for every sequence \((x_n)\) with \(\lVert x_n\rVert\le1\), the sequence
\((Tx_n)\) has a convergent subsequence.

Topic Tags

Bounded Linear Operators Linear Programming and Optimization Sequence Spaces