3
votes
2answers
65 views

Urysohn's lemma with Lipschitz functions

In a complete and separable metric space $(X,\mathrm{d})$ given an open set $U$ and a closed set $K\subset U$. Is it possible to find a Lipschitz function $f$ such that $f|_K=1$ and $f|_{X\setminus ...
2
votes
1answer
53 views

the continuous functions with norm

I'm having trouble trying to understand what does means the first expression in particular the last term in it should we add $\|f\|_{\infty} \leq \infty$ or what i can't see what is his role ...
1
vote
1answer
116 views

Question about Lipschitz function

Suppose $A = (a_{ij})$, $1 \leq i \leq m$, $1 \leq j \leq n$ is an $m \times n$ matrix. then $A: \mathbb{R}^n \to \mathbb{R}^m$ given by $$ A(x) = A(x_1,\dots,x_n) = \left( \sum_{j=1}^n ...
1
vote
2answers
126 views

Proving something is $1$-Lipschitz

(1) Let $(X,d)$ be a metric space, and let A be a non-empty subset. Show that the function $$D_A :X \to [0,\infty ]$$ defined by $$D_A (x) =\inf \{d(x,y) : y \in A\}$$ is $1$-Lipschitz (when ...
3
votes
1answer
730 views

Space of all Lipschitz-continuous functions and the compact and separable sets

Let $C^{0,1}([a,b])$ be the space of all Lipschitz-continuous functions $x\colon [a,b] \to \mathbb{R}$ with the metric $$ d_{0,1}(x,y) := \sup_{a \le t \le b} |x(t) - y(t)| + \sup_{a \le s,t \le b, ...
2
votes
1answer
64 views

Hölderian path connectedness

Let $X$ be a complete metric space and $\alpha \in (0,1)$. Suppose that for every $x,y \in X$ there exists $z \in X$ s.t. $$ d(x,z) \le \frac{1}{2^\alpha}d(x,y), \qquad d(y,z) \le ...