What is the geometrical meaning of the space?

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Consider the Banach space of all lipschitz functions on $X$ such that for each $f$ in the space, \begin{equation*} ||f||=\sup_{x\neq y}\frac{\left\vert f(x)-f(y)\right\vert }{\left\vert x-y\right\vert }. \end{equation*} is a Banach space.