This is an excerpt of page 7 of a paper I am reading:
https://arxiv.org/pdf/1701.07875.pdf
I am confused by the statement, "$\mathcal{W}$ is compact implies that all the functions $f_w$ will be $K$-Lipschitz for some $K$ that only depends on $\mathcal{W}$". Here, we are talking about a family of functions $\{f_w\}_{w \in \mathcal{W}}$. Why does the index coming from a compact space means that the functions will be $K$-Lipschitz continuous? I don't know if this helps, but earlier in the paper, there was a mention that $\mathcal{X}$ is a compact metric space.
I confront same problem with you. Remember that w is weight of neural network, composition of activation functions and linear transformations. Activation functions(sigmoid, Relu, tanh) are 1-Lipschitz function. So Lipschitz constant K of neural network depends on value of w(If it has n multiple layer, roughly saying, it depends on w^n). So if we constrain w to lie in compact space W, closed and bounded space, Lipschitz constant K would be decided.