Prove that \begin{eqnarray} \lim\limits_{p\rightarrow +\infty }||f||_{L^p}=||f||_{L^\infty}. \end{eqnarray}

Asked Mar 26 '18 at 13:50

Active Mar 26 '18 at 13:50

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I am trying to prove this. We are told that $(X,\mathcal{X},\mu)$ is a measure space, and $\mu(X)=1$. We are also told that $f$ is a bounded, measurable function on $X$.

So how can I prove that \begin{eqnarray} \lim\limits_{p\rightarrow +\infty }||f||_{L^p}=||f||_{L^\infty}. \end{eqnarray} Any help please?

asked Mar 26 '18 at 13:50

Francis Daisy

1

Can you prove an inequality? One side $\le$ the other? – GEdgar Mar 26 '18 at 13:56

Prove that \begin{eqnarray} \lim\limits_{p\rightarrow +\infty }||f||_{L^p}=||f||_{L^\infty}. \end{eqnarray}

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