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Possible Duplicate:
$\lim_{n\rightarrow \infty}\int_0^1f_nhdm=\int_0^1fhdm$, prove $f\in L^p(m)$ , where $1\le p<\infty$.

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When ${f_n}$ is defined on [0,1], $ ||f_n||_p\le1$, $1

Prove $f\in L_p$

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    This (essentially unanswered) question is exactly the same -- except that the case $p =1$ is allowed in the other question: http://math.stackexchange.com/q/241076/2012-11-21

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