Implicit function theorem for L^p?

Question

I am studying Ivar Ekeland's book and in one of the Lemma he uses the implicit function theorem on the map \begin{eqnarray} R\times R \times S &\to& L^2(0,T; C^{2n}) \end{eqnarray} where $S$ is the unity sphere in $L^2(0,T; C^{2n})$.

My problem is that the only version I know for the implicit function theorem is for $R^k$ and manifolds.

So, he is thinking of $L^2(0,T; C^{2n})$ as a manifold? Or exist a version of the theorem to $L^p$ spaces?

Answer 1

Banach spaces seem to be the relevant objects here. There is a version of the IFT for Banach spaces. See e.g. this (warning: pdf!)

The relevant stuff start on page $12$.