Saturday, December 10, 2016

real analysis - Prove that limptoinfty|u|p=|u|infty

Let (X,A,μ) be any measure space and let up[1,]Lp(μ). Then lim


I have already proved the inequality \|u\|_{\infty} \leq \liminf_{p\to \infty} \| u \|_p, so I still need to prove \limsup_{p\to \infty} \| u \|_p \leq \| u \|_{\infty}.


Following a hint, I have proved that for any sequence q_n \to \infty one has \|u\|_{p+q_n} \leq \|u\|_{\infty}^{q_n/(p+q_n)} \cdot \|u\|_p^{p/(p+q_n)}.


How can we conclude from this that \limsup_{p\to \infty} \| u \|_p \leq \| u \|_{\infty}?

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