A Frobenius theorem for locally convex global analysis
Let \(E\) and \(F\) locally convex spaces with \(F\) a Banach space. For \(k\in\mathbb{N}\cup \{\infty\}\), consider a \(C^{k+2}\)-manifold modelled on locally convex spaces. A subbundle \(T\) of the tangent bundle \(\pi: TM\to M\) is said to be co-Banach if for each point \(x\in M\), there exists a local \(C^{k+1}\)-trivialization \[ \theta: \pi^{-1}[U]\to U\times E\times F \] over an open neighbourhood \(U\) of \(x\), with \[ \theta[T]= U\times E\times \{0_F\}. \] After having given this definition, the author proves a Frobenius theorem for \(C^{k+2}\)-manifolds modelled on locally convex spaces asserting that if a co-Banach subbundle \(T\) is closed under bracket operation of local \(C^{k+1}\)-sections of \(TM\), then \(T\) is integrable. He based this proof on a simplified version of an implicit function theorem of his recent paper appeared in Stud. Math. 134, No. 3, 235-250 (1999; Zbl 0934.58008)], where the above notion of \(C^k\) is defined. That version has its own interest and its proof is given in the article under review.
\(C^{k+2}\)-manifold modelled on locally convex spaces
scientific article; zbMATH DE number 1423217