In cartesian coordinates, the unit vectors $\{u_x, u_y, u_z\}$ are universal. That is, $u_x(x, y, z)$ is constant and so on for the rest of them. Because of that, the dot product $\langle v | w \rangle$ is defined everywhere in $\mathbb{R}^n$.
But in spherical, $u_\phi(r, \phi, \theta)$ is not constant. It varies. So the dot product $\langle v | w \rangle$ is only defined when $u$ and $v$ share an origin.
But still, I kind of miss that old dot product. Maybe I can get a bilinear form that is defined everywhere in spherical coordinates, by doing something different. What do I need to do to get my universal bilinear form in spherical coordinates?