Re: functional analysis L^p norms

On 1 Sep 2005 08:47:47 -0700, "Volker Boie" <fboie@xxxxxxxxxx> wrote:

>
>sigurgurd_jack@xxxxxxxx schrieb:
>
>> Hi there.
>>
>>
>> Can you help me in showing the following, please?
>>
>>
>> Let (X,f,mu) be a finite measured space and f a measurable real valued
>> function on X.
>>
>> Then lim_{p -> infty} || f ||_p = || f ||_infty.
>>
>>
>> Regards,
>>
>>
>> Jack Sigurgurd

O thought I posted a reply to this yesterday - it seems to
have appeared in the other thread on this topic. I'm so
confused. Anyway:

>I am not sure if this is true if f is not in L^\infty.

It's certainly true whether f is in L^infinity or not.
Hint repeated on request if it isn't visible to you
(see confusion above).

>However, I would
>assume f is a characteristic function on a measurable set, in which
>case your theorem is true. Same in case of simple function f. In
>general approximate f by simple functions in the L\infty norm and
>verify interchanging limits.


************************

David C. Ullrich
.

Relevant Pages
Quantcast