Topology in space of Lipschitz functions.

Question

Let $X$ be a normed space and $L> 0$ be fixed. Consider the set

$$Li_L(X,\mathbb{R})= \{f: X\to \mathbb{R}: |f(x)-f(y)|\leq L\|x-y\|\;\forall \; x,y\in X\}.$$ I am looking for well known topologies on $Li_L.$ Any reference will also be nice.

Answer 1

When considering Lipschitz functions on a normed space $X$, one usually normalizes them by $f(0)= 0$. The space of all normalized Lipschitz functions is called the Lipschitz dual of $X$, denoted $X^\#$. It briefly appears in the book Geometric Nonlinear Functional Analysis by Benyamini and Lindenstrauss. With a search for "Lipschitz dual", "normed space" you'll find more, e.g. this paper.

The space $X^\#$ is a Banach space with the norm $\|f\| = \sup_{a\ne b} |f(a)-f(b)|/\|a-b\|$.

The natural topologies of $X^\#$ are:

• the norm topology,
• the topology of pointwise convergence (which makes closed balls in $X^\#$ compact). It's mentioned in Lipschitz p-summing Operators by Farmer and Johnson.
• the weak* topology, in case it's different from the above (I'm not sure); this is defined because $X^\#$ has a canonical pre-dual constructed by Arens and Eells in On embedding uniform and topological spaces. A more recent paper on this duality is Duality for Lipschitz p-summing operators by Javier Alejandro Chávez-Domínguez.

In principle one could consider the weak topology on $X^\#$ (but what is the dual space of $X^\#$ anyway?) and the topology of uniform convergence, but I haven't seen those.

If you do not normalize $f(0)=0$, the result is just a product of $X^\#$ with $X$.

One more reference: the book Lipschitz Algebras by Nik Weaver, which carefully discusses how to norm and/or normalize Lipschitz functions.