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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

1998 • Question 4

Question

Math 6210 • Real Analysis 1998 • Question 4

Let \(f_n\) be a sequence of functions in \(L^\infty (\mathbb R)\) such that \[ \sup _n\left |\int _{\mathbb R} f(x)g(x)\,dx\right |<\infty \] for all \(g\in L^1(\mathbb R)\). Show that \(\sup _n\lVert f_n\rVert <\infty \).

Raw LaTeX

Let \(f_n\) be a sequence of functions in \(L^\infty(\mathbb R)\) such
that
\[
\sup_n\left|\int_{\mathbb R} f(x)g(x)\,dx\right|<\infty
\]
for all \(g\in L^1(\mathbb R)\). Show that
\(\sup_n\lVert f_n\rVert<\infty\).

Topic Tags

Lebesgue Integration Lp Spaces and Sequence Spaces