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

$$\mathcal{P}=\{f\in L^1 :||f||_{L^1}=1\}\cap L^\infty$$

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.