A sequence $\{f_{n}\}_{n\in I}$ is a frame for a separable Hilbert space $H$ if there exists $0 such that $ A\|f\|^{2} \leq \sum_{n\in I}|\langle f,f_{n}\rangle|^{2}\leq B\|f\|^{2} $ for all $f\in H$.
Some books define a frame for just "Hilbert space" and not mentioning the "separability". Is there any difference between these two cases?