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Let $L^{\infty}$ denote the set of all essentially bounded functions and suppose that $f \in L^{\infty}$ and $g \in L^{\infty}$. Then the product $fg$ is in $L^{\infty}$. Now the question is: prove or disprove: $\|fg\|_\infty \leq \|f\|_\infty \|g\|_\infty$.

Any hints?

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    Ignore the set of measure zero on which f is unbounded, then ignore the set of measure zero on which g is unbounded.2010-11-10

1 Answers 1