Separability of the space of bounded continuous maps - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T13:52:57Z http://mathoverflow.net/feeds/question/53236 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/53236/separability-of-the-space-of-bounded-continuous-maps Separability of the space of bounded continuous maps Orbicular 2011-01-25T14:13:53Z 2011-01-25T14:54:38Z <p>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?</p> <p>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.</p> http://mathoverflow.net/questions/53236/separability-of-the-space-of-bounded-continuous-maps/53237#53237 Answer by BS for Separability of the space of bounded continuous maps BS 2011-01-25T14:39:39Z 2011-01-25T14:54:38Z <p>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 <em>constant</em> maps isn't separable.</p> <p>ADDED : however, there seems to be a an infinite dimensional Sard theorem not requiring separability : </p> <p>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 </p> <p><a href="http://www.ams.org/mathscinet-getitem?mr=322899" rel="nofollow">http://www.ams.org/mathscinet-getitem?mr=322899</a></p>