Definition 2.2. Let X be a metric space. We say that a functional f:X→R is k-Lipschitz continuous if there exists k≥0∈R such that for all x,y∈X,dX(f(x),f(y))≤k⋅|x−y|. In such a case k is called the Lipschitz constant of f.
I think it should be: |f(x)−f(y)|≤k⋅dX(x,y) instead, as f(x),f(y)∈R while x,y∈X.
Found a typo:
I think it should be: |f(x)−f(y)|≤k⋅dX(x,y) instead, as f(x),f(y)∈R while x,y∈X.
This seems correct. I’ll add it to the list of fixes.