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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Spring 2013 • Question 3

Question

Math 6210 • Real Analysis Spring 2013 • Question 3

Let \((X,\mathcal M,\mu )\) be a measure space, \(1<p<\infty \), and \(q=p/(p-1)\). Suppose that \(f_n\in L^p(X)\), \(n=1,2,\ldots \), and that for each \(g\in L^q(X)\), \[ \sup _n \left |\int f_ng\,d\mu \right |<\infty . \] Show that \(\sup _n\lVert f_n\rVert _p<\infty \).

Raw LaTeX

Let \((X,\mathcal M,\mu)\) be a measure space, \(1<p<\infty\), and
\(q=p/(p-1)\). Suppose that \(f_n\in L^p(X)\), \(n=1,2,\ldots\), and that for each
\(g\in L^q(X)\),
\[
\sup_n \left|\int f_ng\,d\mu\right|<\infty.
\]
Show that \(\sup_n\lVert f_n\rVert_p<\infty\).

Topic Tags

Absolute Continuity and Singular Measures Lebesgue Integration Lp Spaces and Sequence Spaces Measure Theory