2

Possible Duplicate:
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 $||f_n ||\rightarrow ||f||$?

Community
  • 1
user58191
  • 457
  • 4
  • 12
  • 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](http://math.stackexchange.com/questions/51502/convergence-a-e-and-of-norms-implies-that-in-lebesgue-space)? – 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

0 Answers0