ca.analysis and odes - Lipschitz functions in $mathbb{R}^n$

Let $f = (f_1,ldots,f_n): [a,b] rightarrow mathbb{R}^n$ be a continuously differentiable function. (See the comments above for an explanation as to why the hypotheses have been strengthened.)

For $1 leq i leq n$, let

$L_i = max_{x in [a,b]} |f_i'(x)|$,

so that, by the Mean Value Theorem, for $x,y in [a,b]$,

$|f_i(x)-f_i(y)| = |f_i'(c)||x-y| leq L_i |x-y|$.

Then, taking the standard Euclidean norm on $mathbb{R}^n$,

$|f(x)-f(y)|^2 = sum_{i=1}^n |f_i(x)-f_i(y)|^2 leq (sum_{i=1}^n L_i^2) |x-y|^2$,

so

$|f(x)-f(y)| leq sqrt{(sum_{i=1}^n L_i^2)} |x-y|$.

Thus we can take

$L = sqrt{sum_{i=1}^n L_i^2}$.

Since all norms on $mathbb{R}^n$ are equivalent -- i.e., differ at most by a multiplicative constant -- the choice of norm on $mathbb{R}^n$ will change the expression of the Lipschitz constant $L$ in terms of the Lipschitz constants $L_i$ of the components, but not whether $f$ is Lipschitz.