Let $f:\mathbb R^n\to \mathbb R^m$ be differentiable and Lipschitz continuous with Lipschitz constant $L$. Then we have $$\|\nabla f(x)\|:=\left( \sum_{i=1}^n\sum_{j=1}^m |\partial_if_j(x)|^2 \right)^{1/2} \leq \sqrt n L.$$
I wonder whether this inequality is optimal or not: Are there any constants $C< \sqrt n$ that satisfies $\|\nabla f(x)\| \leq CL $ ?
The answer depends on which norm you use to define the Lipschitz constant.
For the usual $l^2$ norm, the optimal constant is $\sqrt{\min(n,m)}$. Taking $f_i(x)=x_i$ (with $x_i=0$ for $i>n$) shows that the constant is optimal, as the left side equals $\sqrt{\min(n,m)}$ and the Lipschitz constant is $1$. To see that the inequality is still valid for $m<n$ note that $\sum_{i=1}^n(\partial_i f_j)^2\leq L$ and there are $m$ of these sums on the left side.