Looking for an example for the projection's theorem on an inner product space?

Question

I'm looking for an example of a non-empty, non-convex and complete subset $C$ of an incomplete inner product space $E$ such that if we apply the projection's theorem on $C$ it gives several (maybe infinitely many) solutions or none.

I already have two examples in the case of Hilbert spaces with a non-empty, non-convex and closed (hence complete) subset $C$. Each case is solved depending on the compactness of $C$.

Thanks in advance !

Answer 1

Let $\ell_0$ be the sapce of real sequences with finitely many non-zero terms with the norm induced by $\ell_2$. Let $C=\{e_1,e_2,..\}$. Then $C$ is complete and the distance from $0$ to $C$ is $1$. This distance is achieved at every pint of $C$.