You can define angle in any Hilbert space as the arccos of the inner product over product of norms. I was wondering about doing it in Banach spaces (like $$C[0,1]$$).

Turns out it’s a big mess, like infinite dimensions tend to be.

The paper is https://arxiv.org/abs/1607.06938. There’s also https://arxiv.org/abs/0902.2731 but I didn’t read it yet.