Let $X$ be a (connected, finitely dimensional) topological/smooth/complex manifold and let $i$ be a weakly continuous/continuous/smooth/holomorphic map from $X$ into the dual $F^{*}$ of a real or complex Banach space $F$, which is a homeomorphism onto its image. For $f\in F$ define $f(z)=<f,i(z)>$.
Q1: Is it true that for any $x\in X$ there is a number $n$, neighborhood $U$ of $x$ and $f_1, f_2,...,f_n\in F$, such that the map $y\to[f_1(y), f_2(y),...,f_n(y)]$ is a homeomorphism from $U$ onto its image?
Q2: For the holomorphic case, is it true that if $Y\subset\mathbb{C}^{m}$ is a domain and $j:Y\to F^{*}$ is holomorphic such that $j(Y)\subset i(X)$, then $i^{-1}\circ j$ is holomorphic from $Y$ into $X$?
If Q1 is wrong even in the complex/holomorphic case, there is Q2' for which the hypothesis in Q1 are satisfied (hopefully it's clear what I mean here).
Thank you.
