In the book "Calculus Without Derivatives" written by Jean-Paul Penot there's the following theorem:
THEOREM: Let $X$ and $Z$ be Banach spaces, let $W$ be an open subset of $X$, and let $g:W\to Z$ be a map of class $C^k$ with $k\geq 1$ such that for some $a\in W$ the map $Dg(a)$ is surjective and its kernel $N$ has a topological supplement $M$ in $X$. Then there exist an open neighborhood $U$ of $a$ in $W$ and a diffeomorphism $\varphi$ of class $C^k$ from $U$ onto a neighborhood $V$ of $(0, g(a))$ in $N\times Z$ such that $\varphi (a)=(0,g(a))$ and $g|_U=p\circ \varphi$ where $p$ is the canonical projection from $N\times Z$ onto $Z$.
My question is: in the context of the previous theorem, what does it mean to say that $N:=\ker Dg(a)$ has a topological supplement $M$ in $X$?
Thank you for your attention!