Let $O$ be an open subset of the separable Hilbert space H and $k\geq0$ . Consider $C_b^k(O, Sym(H))$, the space of k-times continuously differentiable maps with values in the bounded symmetric endomorphisms of $H$, bounded up to their k-th derivative. Equipped with the usual norm this space becomes a Banach space. Is this space separable, i.e. has a dense sequence?

I need this result for transversality theory in Morse theory, where the space above serves as a space of suitable perturbations. The separability is needed in order to aplly the Sard-Smale theorem.