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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6420 • PDE

Spring 2017 • Question 1

Question

Math 6420 • PDE Spring 2017 • Question 1

If \((x_n)\) is a weakly convergent sequence in a normed space \(X\), say, \(x_n\to x_0\) weakly, show that there is a sequence \((y_n)\) of linear combinations of elements of \((x_n)\) which converges strongly to \(x_0\). Hint: First prove that \(x_0\in \overline {\operatorname {span}(x_n)}\).

Raw LaTeX

If \((x_n)\) is a weakly convergent sequence in a normed space \(X\), say,
\(x_n\to x_0\) weakly, show that there is a sequence \((y_n)\) of
\underline{linear combinations} of elements of \((x_n)\) which converges strongly to \(x_0\).
Hint: First prove that \(x_0\in\overline{\operatorname{span}(x_n)}\).

Topic Tags

Banach and Normed Spaces Linear Programming and Optimization Sequence Spaces