Let $(X,d)$ be a separable 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?

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?

Let $(X,d)$ be a separable metric space. My question is rather short and to the point: when does the Banach space $C_b(X)$ (with the uniform norm) admit a Schauder basis?

Let $(X,d)$ be a separable 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?