Prove $L_\infty$ is uniform norm

Question

How to prove $L_\infty$ is uniform norm? I.e. $||x||_\infty =\sup_{n}|x_i|=\lim_{r \to \infty} \Big( \sum_{i=1}^n |x_i|^r \Big)^{1/r}$

I don't know how to start, do I need to expand the summation? Any hint is appreciated.

Here: http://math.stackexchange.com/questions/326172/the-l-infty-norm-is-equal-to-the-limit-of-the-lp-norms — ntt, Apr 08 '17 at 08:11

score 2 · Answer 1 · answered Apr 09 '17 at 03:23

For a fixed $n$:

Since $|x_i|=\|x\|_{\infty}$ for at least one $i$, a lower bound for the RHS is $(\|x\|_{\infty}^r)^{1/r}=\|x\|.$

Since $|x_i|\leq \|x\|_{\infty}$ for all $i$, an upper bound for the RHS is $(\sum_{i=1}^n\|x\|_{\infty}^r)^{1/r}=n^{1/r}\|x\|,$

and $\lim_{r\to \infty}n^{1/r}=1.$

1 Answers1