- #1

AxiomOfChoice

- 533

- 1

I just want to make sure I'm straight on the definition.

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!

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!

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