The Lipschitz norm of a function over a domain $D \subseteq \mathbb R^n$ is easy to define: $$\|f\|_{\mathrm{Lip}} = \sup_{x,y \in D} \frac{|f(x) - f(y)|}{|x-y|}.$$ Is there a standard notation for the space of functions whose Lipschitz norm is finite? i.e., $$X(D) = \{f \in C(D) : \|f\|_{\mathrm{Lip}} < \infty \}.$$ Even better, what about functions which are continuously-differentiable and whose derivative has finite Lipschitz norm? $$X^1(D) = \{f \in C^1(D) : \|\nabla f\|_{\mathrm{Lip}} < \infty \}$$ I am using $X$ and $X^1$ here only as placeholders, and would prefer a better notation.

Take the 2-minute tour
×

MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

`$\operatorname{Lip}_1(D)$`

or $\operatorname{Lip}(D)$. – Harald Hanche-Olsen Jul 8 '10 at 20:34