I have been reading about floer homology , and alot of banach (infinite) dimensional manifolds tend to appear. To try and deal with them results such as the inifinite dimensional analogue of the regular value theorem and the sard-smale theorem that applies to fredholm appear.
Now I think to prove the regular value theorem we can just try an use the analogue of the implicit function theorem for banach spaces, but then when we are dealing with fredholm operators I belive we can determine the dimension of the submanifold by the index of the operator.
So my question is if anyone knows a reference that deals with this "classical" manifolds theorem but in infinite dimensions and dealing with tools from Functional analsysis ?
Thanks in advance.
The keyword is "Sard-Smale theorem". The implicit function theorem for Banach manifolds is covered in Lang's book. It is the same statement, all one has to ask is that for all $x$ in $f^{-1}(y)$, we have that $df_x$ is surjective and that the kernel of $df_x: T_x M \to T_{f(x)} N$ be complemented (that is, there is another closed subspace $V$ so that $T_x M = \text{ker}(df_x) \oplus V$).
This assumption is automatically true for Fredholm maps (prove this: if you are going to succeed in this topic you will need to be able to prove small facts like this with ease), and this is also automatically true for arbitrary smooth maps between Hilbert manifolds (you should be able to prove this as well).