Sard-Smale theorem

Statement

Let $M$ and $N$ be [Hilbert manifold]]s and $F$ a Fredholm map from $M$ to $N$. Then:

• If the index of $F$ is negative, then $F^{-1}(p)$ is empty for most <mathp \in N</math>
• If the index of $F$ is nonnegative, then most $p\in N$ are ergular values of $F$ and for those $p$, the fibres are finite-dimensional

This result is closely related to Sard's theorem.