Let $(X,d)$ be a separable and connected metric space. My question is rather short and to the point: do there exist $\{x\_n\}\_{n=0}^{\infty}\subseteq X$ such that
$$
\left\{d(x\_n,\cdot)-d(x\_0,\cdot)\right\}\_{n=1}^{\infty},
$$
is a Schauder basis of $C\_b(X)$? If so, what is this "basis" called in the literature?



When does $C_b(X)$ admit a Schauder Basis?
Note that in order for $C\_b(X)$ to have a Schauder basis, $X$ has to be compact. Indeed, $C\_b(X)$ is naturally isomorphic to $C(\beta X)$ and the latter is non-separable (because $\beta X$ is non-metrisable) [as long as $X$ is a non-compact metric space](https://math.stackexchange.com/a/217422/17929).


Thus, you are in the realm of compact metric space to have a chance for a Schauder basis of some specific form since $C(X)$ always has a Schauder basis for a compact metric connected space. But do you have a basis of the required form already for $C[0,1]$?


