Given a smooth map from $\phi: B \rightarrow M$ where $B$ is a Banach Space and $M$ is a finite dimensional smooth manifold (for example, the end point map for a control system), what is the strongest analogue of Sard's theorem which holds.



Sard's Theorem For Banach Spaces
The classical result is the celebrated Sard-Smale Theorem, see **[1](http://www.jstor.org/stable/2373250?origin=crossref&seq=1#page_scan_tab_contents)**.



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> **Theorem (Smale).** Let $\phi \colon B \longrightarrow M$ be a $C^q$-Fredholm map between separable Banach manifolds, with $q > \max \{0, \, \textrm{index of } \phi\}$. Then the set of regular values of $\phi$ is residual in $M$.
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Here *residual set* means that it is a countable intersection of open dense sets. The Baire Category Theorem implies that a residual set in a complete metric space is dense, in particular its complement contains no open set. 


You cannot expect much better than this, even in your situation. In particular, the Fredholm assumption is an essential one: in fact, Kupka constructed an example of a non-Fredholm, $C^{\infty}$ real function $\phi \colon \ell^2(\mathbb{R}) \longrightarrow \mathbb{R}$ with critical values containing an open set, see **[2](http://www.ams.org/journals/proc/1965-016-05/S0002-9939-1965-0182024-4/home.html)**. 


**References.**


**[1](http://www.jstor.org/stable/2373250?origin=crossref&seq=1#page_scan_tab_contents)** S. Smale: *[An Infinite Dimensional Version of Sard's Theorem](http://www.jstor.org/stable/2373250?origin=crossref&seq=1#page_scan_tab_contents)*, American Journal of Mathematics **87** (1965), 861–866.


**[2](http://www.ams.org/journals/proc/1965-016-05/S0002-9939-1965-0182024-4/home.html)** I. Kupka: *[Counterexample to the Morse-Sard theorem in the case of infinite-dimensional manifolds](http://www.ams.org/journals/proc/1965-016-05/S0002-9939-1965-0182024-4/home.html)*, Proc. Amer. Math. Soc. **16** (1965), 954–957. 


