On injective holomorphic map into a Hilbert Space.

Question

Let $G\subset\mathbb{C}^{n}$ be a domain and let $f$ be an injective holomorphic map from $G$ into a complex Banach space $H$ (you may restrict to Hilbert Spaces only).

Q: Is $f$ necessarily a homeomorphism?

Thank you.

Answer 1

Hint: Start with the algebraic set $A=\{(x,y)\in {\mathbb C}^2: xy=0\}$. Can you express it as the image of the surface $$ S=\{(x,0): x\in {\mathbb C}\} \cup \{(x,1): x\in {\mathbb C}, x\ne 0\} $$ under and injective holomorphic map $F$? If you can, can you also make $S$ smaller so that the smaller surface $S'$ admits a biholomorphic embedding into ${\mathbb C}$ and $F(S)$ is still not a topological manifold?

One can make examples where the domain of your holomorphic map is connected. For this, the image will be contained in, say, a nodal elliptic curve in ${\mathbb C}^2$.