Question

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.

Answer 1

Say $H=L^2(R)$. Then $Sym(H)$ contains $L^\infty(R)$ isometrically (multiplication operators on $H=L^2(R)$), so that even the subspace of constant maps isn't separable.

ADDED : however, there seems to be a an infinite dimensional Sard theorem not requiring separability :

Hausdorff Conullity of Critical Images of Fredholm Maps Frank Quinn and Arthur Sard American Journal of Mathematics Vol. 94, No. 4 (Oct., 1972), pp. 1101-1110

http://www.ams.org/mathscinet-getitem?mr=322899