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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

2005 • Question 5

Question

Math 6210 • Real Analysis 2005 • Question 5

Suppose \(X\) consists of two points \(a\) and \(b\). Define \(\mu (\{a\})=1\), \(\mu (\{b\})=\mu (X)=\infty \), and \(\mu (\emptyset )=0\). For this measure \(\mu \), prove or disprove: \(L^\infty (\mu )\) is the dual space of \(L^1(\mu )\).

Raw LaTeX

Suppose \(X\) consists of two points \(a\) and \(b\). Define
\(\mu(\{a\})=1\), \(\mu(\{b\})=\mu(X)=\infty\), and \(\mu(\emptyset)=0\). For this
measure \(\mu\), prove or disprove: \(L^\infty(\mu)\) is the dual space of
\(L^1(\mu)\).

Topic Tags

Absolute Continuity and Singular Measures Dual Spaces and Hahn-Banach Theorem Lp Spaces and Sequence Spaces Measure Theory