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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Spring 2015 • Question 2

Question

Math 6210 • Real Analysis Spring 2015 • Question 2

Show that if \(f,g\in L^1(\lambda ,\mathbb R)\), then \(\lVert f*g\rVert _1\le \lVert f\rVert _1\lVert g\rVert _1\). Recall \[ (f*g)(x)=\int f(t)g(x-t)\,d\lambda (t). \]

Raw LaTeX

Show that if \(f,g\in L^1(\lambda,\mathbb R)\), then
\(\lVert f*g\rVert_1\le \lVert f\rVert_1\lVert g\rVert_1\). Recall
\[
(f*g)(x)=\int f(t)g(x-t)\,d\lambda(t).
\]

Topic Tags

Lebesgue Integration Lp Spaces and Sequence Spaces