This is a follow-up of a question of mine with a similar title. I am interested in Morse homology (on Hilbert manifolds), more specifically with "generic" perturbations of the metric tensor (under the heading of "transversality"). The space of perturbations to use should have the property of separability in order to apply the Sard-Smale theorem. 


Now comes the question, which is supposed to help me in these matters: Let H be a separable Hilbert space and let B be the closed unit ball in H. Is the space C^b(B), the space of continuous bounded functions on the closed unit ball endowed with the sup-norm, a separable space?


The previous question of mine replaced B above with an open subset of the Hilbert space. Then the answer turns out to be NO. Notice that if H is finite dimensional, the answer is YES.



separability of a certain space of continuous functions, II
**No.** Ady's construction still works, I think. Here's another, easier one:


Choose an orthonormal basis $\{e\_{n}\}$ of $H$. Since $\|e\_{i} - e\_{j}\| = \sqrt{2}$, the continuous functions $f\_{n}(x) = \max\{0, 1 - 2 \cdot \|e\_{n} - x\|\}$ have disjoint support and sup-norm $1$. This gives an isometric embedding $\ell^{\infty} \to C\_{b}(B)$ by sending a bounded sequence $(a\_{n})$ to $\sum a\_{n} f\_{n}$.


A simple modification of this argument shows that $C\_{b}(X)$ contains a copy of $\ell^{\infty}$ (and thus isn't separable) whenever the completely regular space $X$ has a countable discrete subset.


