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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Spring 2009 • Question 2

Question

Math 6210 • Real Analysis Spring 2009 • Question 2

Let \((X,\mu )\) be a measure space with \(\mu (X)=1\), and let \(f\in L^\infty (X)\). Prove that \(\lVert f\rVert _p\to \lVert f\rVert _\infty \) where \(\lVert f\rVert _p\) denotes the \(L^p\) norm of \(f\).

Raw LaTeX

Let \((X,\mu)\) be a measure space with \(\mu(X)=1\), and let
\(f\in L^\infty(X)\). Prove that \(\lVert f\rVert_p\to\lVert f\rVert_\infty\)
where \(\lVert f\rVert_p\) denotes the \(L^p\) norm of \(f\).

Topic Tags

Absolute Continuity and Singular Measures Lp Spaces and Sequence Spaces Measure Theory Outer Measures and Caratheodory Measurability