sequence of functions in $L^p$

Asked Jan 19 '13 at 18:47

Active Jan 19 '13 at 18:47

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Convergence a.e. and of norms implies that in Lebesgue space

Let $\lbrace f_n \rbrace$ be a sequence of functions in $L^p$ ($1 \leq p < \infty)$ that converge almost everywhere to a function $f$ in $L^p$.

What's the shortest proof showing that $f_n \rightarrow f$ in $L^p$ if and only if $||f_n ||\rightarrow ||f||$?

edited Apr 13 '17 at 12:21

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asked Jan 19 '13 at 18:47

user58191

Do you have a long proof you are trying to shorten? Are both directions too long? – Jonas Meyer Jan 19 '13 at 18:51
In particular, proving convergence in $L^p$ is longer than what I expect. The other direction is a cinch.
I'm trying to shorten proofs as I'm reviewing for exams, and looking for shorter proofs has helped me remember stuff.
– user58191 Jan 19 '13 at 19:03
How about this? – David Mitra Jan 19 '13 at 19:10
wow! That's perfect. I didn't see that though when I was searched this site. Thanks. – user58191 Jan 19 '13 at 19:18

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