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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

2004 • Question 4

Question

Math 6210 • Real Analysis 2004 • Question 4

Let \(f_1,f_2,\cdots ,f_n,\cdots \) be a sequence of functions in \(L^1(\mathbb R)\) such that \[ \lim _{n\to \infty }\int _{\mathbb R} f_n(x)g(x)\,dx=0 \] for every \(g\in L^\infty (\mathbb R)\). Show that \(f_1,f_2,\cdots ,f_n,\cdots \) is a bounded sequence in \(L^1(\mathbb R)\).

Raw LaTeX

Let \(f_1,f_2,\cdots,f_n,\cdots\) be a sequence of functions in
\(L^1(\mathbb R)\) such that
\[
\lim_{n\to\infty}\int_{\mathbb R} f_n(x)g(x)\,dx=0
\]
for every \(g\in L^\infty(\mathbb R)\). Show that
\(f_1,f_2,\cdots,f_n,\cdots\) is a bounded sequence in \(L^1(\mathbb R)\).

Topic Tags

Almost Everywhere and Convergence in Measure Lebesgue Integration Lp Spaces and Sequence Spaces