I just want to make sure I'm straight on the definition.(adsbygoogle = window.adsbygoogle || []).push({});

Am I correct in assuming that, if I want to show that a sequence [itex]\langle f_n \rangle[/itex] of functions converges to 0 in the [itex]L^1[/itex] norm, I have to show that, for every [itex]\epsilon > 0[/itex], there exists [itex]N \in \mathbb N[/itex] such that

[tex]

\int |f_n| < \epsilon

[/tex]

whenever [itex]n > N[/itex]?

Also, is it possible for a sequence of functions to converge uniformly to 0 and yet *not* converge to 0 in the [itex]L^1[/itex] norm? (I'm pretty sure I have an example of this if the above definition is correct.)

Thanks!

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Convergence of a sequence of functions to zero in the L1 norm?

**Physics Forums | Science Articles, Homework Help, Discussion**