Re: When does weak convergence imply strong convergence?
- From: David C. Ullrich <ullrich@xxxxxxxxxxxxxxxx>
- Date: Wed, 04 May 2005 10:27:28 -0500
On Wed, 04 May 2005 00:06:09 GMT, Kira Yamato <no@xxxxxxxx> wrote:
>Suppose f_n -> f weakly in whatever space E. Then if
>(1) |f_n|_E -> |f|_E
>also, then f_n -> f strongly in that space.
Not so (well, depending on what "whatever" means).
If E is L^infinity and f_n is the characteristic
function of (-infinity, n) then f_n -> 1 weakly,
|f_n| = 1, but f_n does not converge to 1 in norm.
>Condition (1) is a sufficient condition.
>
>My question is this: Is (1) necessary?
>
>-Kira
************************
David C. Ullrich
.
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