Let $M,N$ be two Banach or Hilbert manifolds. Assume that $P\subset N$ is a smooth submanifold of $N$. We say that a smooth map $f:M \to N$ is Fredholm transversal to $P$ if for every $x\in M$ with $f(x)\in P$ the pair$(Df_x(T_x M), T_{f(x)} P)$ would be a Fredholm pair. (Recall that a Fredholm pair in a Banach or Hilbert space H is a pair of closed subspaces $(V,W)$ such that $V\cap W$ is a finite dimensional space and $V+W$ is a closed subspace of $H$ with finite codimension.
Is this terminology introduced in some reference? Let $H$ be a Hilbert space. Is there a submanifold $P\subset H$ for which there is no a surjective map $f:H \to H$ such that $f$ is Fredholm transverse to $P$? In particular does the unit sphere $P$ of $H$ admit a surjective Fredholm transversal map?
