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if $\| h.\psi \|_{2}<\infty$ for all $h\in L_{2}[a,b]$ then $\|\psi\|_{\infty}<\infty$ ($\|\psi\|_{\infty}$ to be essential supremum of $\psi$ ).

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What is the question here? –  Thomas E. Feb 3 '13 at 9:26
prove that Essential supremum of the function ψ is finite. –  Alexander Osorio Feb 3 '13 at 9:31
Why do you think it is true? –  Michael Greinecker Feb 3 '13 at 9:40
Asking it once here is enough. –  Michael Greinecker Feb 3 '13 at 9:42

1 Answer 1

The statement is not true.

Either using Baire's theorem or finding an explicit example, we can show that "$\lVert h\phi\rVert_{L^2}<\infty$ for all $h\in L^2$ is equivalent to $\phi\in L^2$. Now just find a function in $L^2$ which is not in $L^\infty$.

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I already did by contradiction –  Alexander Osorio Feb 4 '13 at 9:35

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