Skip to content

Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6420 • PDE

Spring 2015 • Question 2

Question

Math 6420 • PDE Spring 2015 • Question 2

Let \((x_n)\) be a sequence in a Hilbert space \(H\). Show that: \[ x_n\to x\ \textbf {strongly} \quad \text {if and only if}\quad (x_n\to x\ \textbf {weakly}\text { and }\lVert x_n\rVert \to \lVert x\rVert ). \]

Raw LaTeX

Let \((x_n)\) be a sequence in a Hilbert space \(H\). Show that:
\[
  x_n\to x\ \textbf{strongly}
  \quad\text{if and only if}\quad
  (x_n\to x\ \textbf{weakly}\text{ and }\lVert x_n\rVert\to\lVert x\rVert).
\]

Topic Tags

Hilbert Spaces and Orthogonality Linear Programming and Optimization Sequence Spaces