A sequence $\{f_{n}\}_{n\in I}$ is a frame for a separable Hilbert space $H$ if there exists $0<A\leq B<\infty$ 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?
If the Hilbert space is not separable and $\{f_n\}_{n\in I}$ is a frame in this sense, then take $f\neq 0$ orthogonal to all the $f_n$'s: it's possible, otherwise the Hilbert space would be separable. By the first inequality, we would have that $f=0$, which is not possible.
But it can make sense if we deal with an arbitrary set. For example, take $\ell²(0,1)$, which is not separable and $f_i(k):=\delta_{ik}$ for $i,k\in (0,1)$.