1-Let $(\mathcal{X},\|.\|_q)$ be a compact metrix space. Also let $f(\mathbf{x})=A\mathbf{x}$, $\forall \mathbf{x}\in\mathcal{X}$. Can, I define the Lipschitz continuity constraint on $f(.)$ w.r.t vector norms, i.e., \begin{equation} \|f(\mathbf{x}_1)-f(\mathbf{x}_2)\|_p\leq L\|\mathbf{x}_1-\mathbf{x_2}\|_q \end{equation}
2 - Now, let $f(\mathbf{x})=A\mathbf{x}+\mathbf{n}$, where $\mathbf{n}$ be a bounded random noise. Can the same Lipschitz constraint be applied to this $f(.)$ as well? To the best of my knowledge, Lipschitz continuity constraint cant be applied to random signals.