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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Fall 2013 • Question 1

Question

Math 6210 • Real Analysis Fall 2013 • Question 1

Let \((X,M,\mu )\) be a measure space, and suppose that \(1\le p<q<r\). Consider the vector space \(L^p\cap L^r\), with norm \(\lVert f\rVert =\lVert f\rVert _p+\lVert f\rVert _r\). Show that \(L^q\subset L^p\cap L^r\), and that the inclusion map \(i:L^q\to L^p\cap L^r\) is continuous.

Raw LaTeX

Let \((X,M,\mu)\) be a measure space, and suppose that
\(1\le p<q<r\). Consider the vector space \(L^p\cap L^r\), with norm
\(\lVert f\rVert=\lVert f\rVert_p+\lVert f\rVert_r\). Show that
\(L^q\subset L^p\cap L^r\), and that the inclusion map
\(i:L^q\to L^p\cap L^r\) is continuous.

Topic Tags

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