A rank theorem for infinite dimensional spaces

Abstract:

Suppose $X$ is a Banach space, $U$ is an open set of $X$ containing 0, and $f$ is a continuously differentiable function from $U$ into $X$ satisfying $f(0) = 0$ and $f’{(0)^2} = f’(0)$. An additional hypothesis is given for $f$ which, in case $X$ is finite dimensional, is equivalent to assuming rank $f’(x) = \operatorname {rank} f’(0)$ for all $x$ in some neighborhood of 0. Under this hypothesis one obtains a local factorization of $f$ into ${h_1} \circ f’(0) \circ {h_2}$ where each of ${h_1}$ and ${h_2}$ is a continuously differentiable homeomorphism. In addition there is a neighborhood of 0 in ${f^{ - 1}}(\{ 0\} )$ which is the image of a continuously differentiable retraction. An application of these results to the theory of differentiable multiplications is given.