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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Summer 2018 • Question 5

Question

Math 6210 • Real Analysis Summer 2018 • Question 5

Let \(X=[0,1]\) with the usual measure. Let \(p\ge 1\). For \(n=1,2,\ldots \), let \[ f_n=n^{1/p}\cdot \chi _{[0,1/n]}. \] Prove that the sequence \(f_n\) converges weakly to \(0\) in \(L^p([0,1])\) if and only if \(p>1\). Use that \[ L^p([0,1])^*\cong L^q([0,1]). \]

Raw LaTeX

Let \(X=[0,1]\) with the usual measure. Let \(p\ge1\). For
\(n=1,2,\ldots\), let
\[
f_n=n^{1/p}\cdot \chi_{[0,1/n]}.
\]
Prove that the sequence \(f_n\) converges weakly to \(0\) in \(L^p([0,1])\) if and
only if \(p>1\). Use that
\[
L^p([0,1])^*\cong L^q([0,1]).
\]

Topic Tags

Absolute Continuity and Singular Measures Lp Spaces and Sequence Spaces Measure Theory Weak Topologies and Weak Convergence