Lipschitz Map Between Metric Spaces

Let $(X,d_X)$ and $(Y,d_Y)$ be metric spaces and let $\Lambda$ be a nonnegative real number.

A map $f:X\to Y$ is Lipschitz with constant $\Lambda$ if

$$d_Y\big(f(x),f(x')\big)\le\Lambda\,d_X(x,x')\qquad\text{for all }x,x'\in X,$$

and Lipschitz if it is Lipschitz with constant $\Lambda$ for some nonnegative real number $\Lambda$.

When $X$ is a nonempty subset of Euclidean space $\mathbb{R}^n$ and $Y$ is a nonempty subset of $\mathbb{R}^k$, the metrics are understood to be the restrictions of the Euclidean distances unless another choice is stated.