Given contraction in $\|\cdot \|_2$ norm, what is the contraction in $\|\cdot\|_\infty$ norm?

Question

Suppose that I have a contraction $f$ with respect to $\|\cdot\|_2$

$$\|f(x) - f(y)\|_2 \leq l\|x-y\|_2, l\in[0,1)$$

What can I say about $k>0$

$$\|f(x) - f(y)\|_\infty \leq k\|x-y\|_\infty$$

What about any $1\leq p \leq \infty$ norm?

$$\|f(x) - f(y)\|_p \leq k\|x-y\|_p$$

Answer 1

Two norms on $\mathbb{R}^n$ are equivalent: there exist $c_p,C_p>0$ such that

$\forall x\in\mathbb{R}^n\quad c_p\|x\|_p\leq\|x\|_2\leq C_p\|x\|_p$.

This yields $c_p\|f(x)-f(y)\|_p\leq lC_p\|x-y\|_p$ and therefore $\|f(x)-f(y)\|_p\leq l\frac{C_p}{c_p}\|x-y\|_p$.