Euclidean norm proof

Question

I was wondering whether anyone knew the proof of the Euclidean norm $ \|\cdot\|_2 \leq K \|\cdot\|_\infty$ for a real number $K$.

Any help will be appreciated

Answer 1

Suppose I know the $\ell^\infty$ norm of my grocery store is $d$ miles. (Naturally I take myself to be the center of the universe). Then at worst the store is, say, $d$ miles north and $d$ miles east of me, which is a total distance of $\sqrt{d^2 + d^2} = d \sqrt{2}$ miles as the crow flies. So the statement holds taking $K = \sqrt{2}$.

Now just generalize to more dimensions.

Answer 2

By definition

$$\begin{align} \|x\|_2=&\sqrt{\sum_{i=0}^nx_i^2}\\\|x\|_{\infty}=&\sup_{i}|x_i|\end{align}$$

Now one has

$$x_i^2=|x_i|^2\leq\left(\sup_{i}|x_i|\right)^2=\|x\|_{\infty}^2$$

Adding the inequalities from $i=1$ to $i=n$ and taking the square root (all the terms are non negative) one gets

$$\|x\|_{2}\leq\sqrt{n}\|x\|_{\infty}$$