Re: cl(bigcap_{1<=q<=infty} L^q) = L^p

On Thu, 26 May 2005 14:01:43 GMT, rob@xxxxxxxxxxxxxx (Rob Johnson)
wrote:

>In article <6hjle.8288$uR4.7524@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
>Kira Yamato <no@xxxxxxxx> wrote:
>>Ok, I'm trying to verify my often flawed logic, in this case, about
>>density of L^p(R) spaces.
>>
>>Let f in L^p(R). Define the n-boxed function of f,
>> f_n(x) = 0 if |x|>n,
>> = n if f(x)>n,
>> = -n if f(x)<n,
>> = f(x) otherwise.
>>Now, {f_n}, being bounded in domain and range, are clearly in L^q(R) for
>>all 1<=q<=infty.
>>
>>Since clearly
>> |f-f_n| <= |f|,
>>Lebesgue's Dominated Convergence Theorem gives
>> lim_n \int |f-f_n| = \int lim_n |f-f_n| = 0.
>>So,
>> f_n -> f strongly in L^p.
>>
>>But {f_n} are also in every other L^q spaces. So, we conclude that
>> bigcap_{1<=q<=infty} L^q
>>is dense in L^p.
>>
>>Is this correct or should I retake Real Variable I? Thanks ahead for
>>helping me.
>
>The conclusion in the text of your message is correct. However, the
>equality in the title of your post is not. Certainly, L^p is in the
>closure you specify, but it is not all of that closure. Therefore,
>the equality is not warranted.

I assumed that "closure" must mean "closure in the L^p norm".
What topology are you assuming?

>Rob Johnson <rob@xxxxxxxxxxxxxx>
>take out the trash before replying


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

David C. Ullrich
.

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