Sard–Smale theorem

Remarks

The theorem is due to Smale and extends Sard's theorem to infinite dimension. Measure zero has no meaning in an infinite-dimensional Banach space, so it is replaced by the Baire notion of a residual set. The proof is local. Near a point $x$, the Fredholm property lets one write $f$ in charts as $(u, v) \mapsto (u, g(u, v))$, with $u$ in a Banach space and $g$ with values in a finite-dimensional space. The critical values near $x$ are then those of finite-dimensional maps, which have measure zero by Sard's theorem. Fredholm maps are locally proper, so the critical values coming from a small closed neighbourhood form a closed set with empty interior. Second countability gives countably many such pieces.

At a regular value $y$, the implicit function theorem shows that $f^{-1}(y)$ is a $C^k$ submanifold of $M$ of dimension $\operatorname{ind} f$, empty if the index is negative. Example: if $f \colon M \to N$ is a smooth map between finite-dimensional manifolds, it is Fredholm of index $\dim M - \dim N$ and the theorem reduces to Sard's theorem, with residual in place of measure zero.

Going deeper

For a student

The theorem is the basic transversality tool of global analysis. Take a nonlinear elliptic equation, written as $f(x) = y$ with $f$ Fredholm. The theorem shows that for generic data $y$ the solution set is a smooth finite-dimensional manifold. One can also apply it to the projection onto the parameter space of the zero set of a universal family $F(x, \lambda)$, where $\lambda$ is a metric or a perturbation. This gives the parametric transversality used for moduli spaces of instantons, of pseudoholomorphic curves and of Morse flow lines.