Is the spherical coordinate system a vector space?

Question

There are basis in spherical coordinate system but I think it does not satisfy vector axioms. Is calling them basis an abuse of notation? Edit:Look in the comments of accepted answer for the complete answer.

Answer 1

For any point $P$ not on the $z$-axis, the vector triple $$ \bigl( \mathbf{e}_r(P),\mathbf{e}_\theta(P),\mathbf{e}_\phi(P) \bigr) $$ indeed is an ON-basis for $\mathbf{R}^3$. (Or maybe rather for the tangent space to $\mathbf{R}^3$ at the point $P$, if that distinction makes any sense to you.)

But that doesn't mean that the spherical coordinate system is a vector space. What would it even mean for a coordinate system to be a vector space?