Kuratowski embedding

Remarks

By the triangle inequality, $\abs{d(x, y) - d(x', y)} \le d(x, x')$ for every $y$. With $x' = x_0$, this shows that $\Phi(x)$ is bounded by $d(x, x_0)$. In general, it gives $\norm{\Phi(x) - \Phi(x')}_\infty \le d(x, x')$. Taking $y = x'$ gives the value $d(x, x')$, so equality holds.

If $X$ is bounded, the simpler map $x \mapsto d(x, \cdot)$ already works. If $X$ is separable, the same idea with a countable dense subset embeds $X$ isometrically in the sequence space $\ell^\infty$.

Going deeper

For a student

Since $\ell^\infty(X)$ is complete, the closure of $\Phi(X)$ is a complete metric space containing $X$ densely. This gives a quick construction of the completion of a metric space. The embedding also lets one take convex hulls and straight segments of points of $X$ inside a Banach space. For compact $X$ and $Y$, it shows that the distance on the Gromov–Hausdorff space can be computed inside the single space $Z = \ell^\infty$.