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This is a functional analysis qualifying exam problem that I can't figure out. Any assistance would be appreciated since I have to take a similar qual soon. I was able to make some limited progress in the p=2 case using Holders inequality.

Suppose [itex]f_n, f\in L^p[/itex] where [itex]1\le p <\infty[/itex] and that [itex] f_n \rightarrow f[/itex] a.e. Show that [itex]\|f_n-f\|_p \rightarrow 0[/itex] iff [itex] \|f_n\|_p \rightarrow \|f\|_p [/itex].

Suppose [itex]f_n, f\in L^p[/itex] where [itex]1\le p <\infty[/itex] and that [itex] f_n \rightarrow f[/itex] a.e. Show that [itex]\|f_n-f\|_p \rightarrow 0[/itex] iff [itex] \|f_n\|_p \rightarrow \|f\|_p [/itex].

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