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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Fall 2017 • Question 3

Question

Math 6210 • Real Analysis Fall 2017 • Question 3

Show that if \(v_i\in \mathcal H\) and \(v_i\) converges weakly to \(u\), then \[ \liminf _{i\to \infty }\lVert v_i\rVert \ge \lVert u\rVert , \] and that equality occurs only when there exists a subsequence \(v_{n_j}\) so that \(v_{n_j}\) converges to \(u\) in norm.

Raw LaTeX

Show that if \(v_i\in\mathcal H\) and \(v_i\) converges weakly to
\(u\), then
\[
\liminf_{i\to\infty}\lVert v_i\rVert\ge\lVert u\rVert,
\]
and that equality occurs only when there exists a subsequence \(v_{n_j}\) so
that \(v_{n_j}\) converges to \(u\) in norm.

Topic Tags

Weak Topologies and Weak Convergence