[Math] Vector valued Mean value theorem: Norm for the gradient

Question

The wikipedia article on the vector valued Mean value theorem, says

For $f:\mathbb R^n \to \mathbb R^n$, if the gradient is bounded,
$$
\| \nabla f \| \le M,
$$
then
$$
\|f(x)-f(y) \| \le M \|x-y\|.
$$

What is the norm used for the gradient $\| \nabla f \|$?

I tried to look in some other references.
There the matrix-norm is mentionned.
They gave one example, where I don't understand how to get the bound of $\frac14$ on $\|\nabla f \|$.

Define $g:\mathbb R_+^2 \to \mathbb R_+^2$ with
$$
g(x,y)=\left( \frac{1}{4+x+y}, \frac{1}{4+x+y} \right),
$$
then, entry-wise,
$$
\nabla f \le \begin{pmatrix} \frac{1}{16} & \frac{1}{16} \\ \frac{1}{16} & \frac{1}{16} \end{pmatrix}=A,
$$
whence
$$
\|\nabla f \| \le \|A\|=\frac14.
$$

what norm reduces $\|A\|$ to $\frac14$?

Answer 1

For finite-dimensional spaces, matrix norms are equivalent, so the choice does not matter.