I have a quick question about the proper way of denoting the function space of continuous functions defined on a close interval. for example the function space of Legendre Polynomial. I think it probably looks like $C({\mathbb{R}(a,b)})$, but couldn't be sure, could you help me with it?
Also a related question, how then to denote the function space defined by $C({\mathbb{S}^3}) \times C({\mathbb{R}(a,b)})$, basically the function space of all continuous functions defined within a 3D sphere of certain diameter?
(I assumed $C({\mathbb{S}^3})$ is the space of continuous functions defined on a spherical surface, basically thespace spanned by spherical harmonics, please let me know if I'm using it wrong).
Thank you very much!