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Sard--Smale residual regular values for Fredholm maps

Statement

Let $P : X \to Y$ be a $C^{h}$ Fredholm map of index $m$ between countable-base Banach manifolds. If $h > \max {m, 0}$ , its regular values form a residual subset of $Y$ .

Facts & Assumptions

Given: Such a $C^{h}$ Fredholm map $P$ with $h > \max {m, 0}$ .

[F1]

Fredholmness means finite kernel and cokernel, closed range, and locally constant index (Fredholm maps and regular values on countable-base Banach manifolds).

[F2]

Residual means complement of a meagre set (Nowhere dense, meagre, residual, and comeagre subsets of a topological space).

[F3]

The finite-dimensional Sard theorem gives null critical values at its stated differentiability threshold (Morse-Sard for Euclidean maps).

Proof

technique · direct

1.1

At each $x$ , the finite-dimensional kernel and cokernel in [F1] permit the standard local Lyapunov--Schmidt reduction of $P$ to a $C^{h}$ map between finite-dimensional spaces whose source-target dimension difference is $m$ .

F1given

2.1

Fredholm maps are locally proper on suitable closed neighbourhoods. Using the countable bases, choose countably many such neighbourhoods $N_{i}$ covering $X$ . On each $N_{i}$ , Lyapunov--Schmidt reduction and [F3] show that the image of the critical set has empty interior; properness makes that image closed, hence nowhere dense.

F3step 1.1algebra

3.1

Every critical value belongs to one of these countably many nowhere-dense images. Their union is meagre, so [F2] makes its complement residual; every point of that complement is a regular value of $P$ .

F2step 2.1∎

Depends on

Used by

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Sources

Stephen Smale, An Infinite Dimensional Version of Sard's Theorem, Theorem (1.3) (standard reference, not scraped)