$L^{\infty}$ norm

Question

For Lebesgue $p$-integrable functions, what would be the formula for $$\left(\int_0^1 \sum_{i=1}^n | f_i(x)|^p dx\right)^{\frac{1}{p}} $$ as $p\to +\infty$? Would it be $$\max_i \sup_{[0,1]} |f_i(x)|,$$ or $$\sum_{i=1}^n \sup_{[0,1]} |f_i(x)|?$$

Answer 1

Denote by $X_p\in\mathbb R^n$ the vector $(\Vert f_1\Vert_{L_p(0,1)},\dots ,\Vert f_n\Vert_{L_p(0,1)} )$. Then the quantity you are looking at, say $I_p$, is equal to $\Vert X_p\Vert_{\ell_p}$.

As $p\to\infty$, you know that $X_p\to X=(\Vert f_1\Vert_\infty,\dots ,\Vert f_n\Vert_\infty)$ component-wise, hence for any norm on $\mathbb R^n$; and you also know that $\Vert X\Vert_{\ell_p}\to \Vert X\Vert_\infty=\max( \Vert f_1\Vert_\infty,\dots ,\Vert f_n\Vert_\infty)$.

Since $ I_p=\Vert X\Vert_{\ell_p}+ (\Vert X_p\Vert_{\ell_p}-\Vert X\Vert_{\ell_p})$ and $\vert \Vert X_p\Vert_{\ell_p}-\Vert X\Vert_{\ell_p} \vert\leq \Vert X_p-X\Vert_{\ell_p}\leq \Vert X_p-X\Vert_{\ell_1}\to 0$, it follows that $I_p\to \Vert X\Vert_\infty$. So the limit is $\max\limits_i\; \sup\limits_{[0,1]}\vert f_i(x)\vert $.