Surjectivity alone does not imply a complemented kernel

The regular value theorem of this page (Regular value theorem for Banach manifolds) separately requires the specified atlas of the domain manifold to be maximal and assumes that at every point of the level set the derivative is surjective with complemented kernel (A complemented closed subspace of a normed space). For general Banach spaces the second clause is not a consequence of the first, and it cannot be dropped:

• Equivalent formulation. Assume the Axiom of Dependent Choice (The axiom of dependent choice: a relation in which every element is related to something admits an $\mathbb{N}$-indexed chain). For a surjective bounded linear operator $L:X\to Y$ between Banach spaces, $\ker L$ is complemented in $X$ if and only if $L$ admits a bounded right inverse (Under Dependent Choice, a surjective bounded operator between Banach spaces has a bounded right inverse exactly when its kernel is complemented). Thus the hypothesis of the regular value theorem is exactly the requirement that the derivative admit a bounded right inverse along the level set.

• Why surjectivity by itself is not enough, and where the failure is seen. The open mapping theorem makes $L$ open, but it does not supply a bounded linear right inverse. Without one, the fibres of the linear map are affine translates of a closed uncomplemented subspace and cannot be the split coordinate slices demanded by Split Banach submanifold. The companion page carries the standard witness: $c_0$ is a closed subspace of ${\mathrm{\ell }}^{\mathrm{\infty }}$ that is not complemented in it, so the identity chart has no split-coordinate decomposition for the pair $({\mathrm{\ell }}^{\mathrm{\infty }},c_0)$. Because ${\mathrm{\ell }}^{\mathrm{\infty }}$ is not second countable, this is a Banach-space obstruction and not a counterexample involving a Banach manifold under this library's convention. It shows why complementability is a genuine extra linear hypothesis; it does not by itself exhibit a regular level set in the manifold category.

• Automatic cases, and the ones that matter below. A closed subspace that is finite dimensional or of finite codimension is automatically complemented (Finite-dimensional subspaces are complemented, Closed finite-codimensional subspaces are complemented). In particular, if $L$ is a Fredholm operator (Fredholm operator cokernel and index) or if its target is finite dimensional, then surjectivity of $L$ implies that $\ker L$ is complemented, so the extra clause of the theorem is automatic in those cases. This is why the Fredholm-and-transversality results of this page can print the complemented-kernel hypothesis once and use it everywhere without further case distinctions, while the abstract theorem states it outright.

• Bookkeeping. The equivalence in the first bullet is a Dependent Choice theorem, and the regular value theorem is proved under the Axiom of Choice (The Axiom of Choice), which supplies DC; it also has the independent structural hypothesis that the specified domain atlas is maximal. The counterexample on the companion page uses only the Axiom of Countable Choice, since the non-complementation it appeals to is proved at that strength.