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  "body": "\n## Statement\n\nLet $M$ and $N$ be [[Banach manifolds|9K15AMGBV9]], with $M$ connected and second countable.\nLet $f \\colon M \\to N$ be a [[Fredholm map|GFXQ6HEM65]] of class $C^k$ with index $\\operatorname{ind} f$, where $k \u003e \\max(\\operatorname{ind} f, 0)$.\nA point $y \\in N$ is a *regular value* of $f$ if $d f_x$ is surjective for every $x \\in f^{-1}(y)$.\nThen the set of regular values of $f$ is a [[residual set|GRH8MCVN3N]] in $N$, that is, it contains a countable intersection of open dense sets.\n\n## Remarks\n\nThe theorem is due to Smale and extends [[Sard's theorem|Z7R8J71SJN]] to infinite dimension.\nMeasure zero has no meaning in an infinite-dimensional Banach space, so it is replaced by the Baire notion of a residual set.\nThe proof is local.\nNear 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.\nThe critical values near $x$ are then those of finite-dimensional maps, which have measure zero by Sard's theorem.\nFredholm maps are locally proper, so the critical values coming from a small closed neighbourhood form a closed set with empty interior.\nSecond countability gives countably many such pieces.\n\nAt 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.\nExample: 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.\n\n## Explanations\n\n### L1 human\n\nSard's theorem says that almost every value of a smooth map is regular, so that its preimage is a nice manifold.\nIn infinite dimension there is no good notion of volume.\nFor Fredholm maps one can still say that regular values are generic: they form a set that is as large as possible in the sense of Baire.\n\n### L2 human\n\nThe theorem is the basic transversality tool of global analysis.\nTake a nonlinear elliptic equation, written as $f(x) = y$ with $f$ Fredholm.\nThe theorem shows that for generic data $y$ the solution set is a smooth finite-dimensional manifold.\nOne 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.\nThis gives the parametric transversality used for moduli spaces of instantons, of pseudoholomorphic curves and of Morse flow lines.\n\n",
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