What would be the correct way to prove a set of basis functions spans the rotation-invariant subspace of R3? I have a set of function that I know spans R3, and by combining the projection results in a specific way, I found a this combination to be rotation-invariant and I have a way to prove it. However, I'm not sure how to mathematically prove/show it's complete in the rotation-invariant subspace, or what criteria needs to be met to claim such completeness.
Maybe this is not the correct way to ask it, and we need to rephrase it somehow, but such basis of rotation-invariant subspace would have the same projection/representation for functions that are rotation transformations of each other, yet different for different functions.
Maybe we can rephrase it as: proving the completeness for functions in R3 but with their internal reference frame? But I'm not sure how to show that either... and I feel like this would involve some alignment issue.