Is the Lebesgue-Integral Continuous with Respect to the L^\infty Norm?

[Pere Callahan]

Hi,

I was wondering if the Lebesgue-integral is continuos with respect to the L^\infty norm.

More precisely, assume there is a space of functions

<br /> \mathcal{P}=\{f\in L^1 :||f||_{L^1}=1\}\cap L^\infty<br />

endowed with the essential supremum norm ||\cdot||_{L^\infty}. If there is then a Cauchy sequence f_1,f_2,\dots in \mathcal{P}, can one conclude that the limit (which exists in L^\infty) is also integrable with L^1 norm one?

Thank you very much.
Pere

 

[morphism]

How about something like f_n = \frac{1}{n} \chi_{[0,n]} in L^1(\mathbb{R})? Here we have \|f_n \|_{L^1} = 1 and \|f_n - 0\|_{L^\infty} = \frac{1}{n} \to 0, while \| 0 \|_{L^1} = 0.

 

[Pere Callahan]

Thanks morphism, that seems to answer my question.