Let $T=V/L$ be a complex tori with lattice $L$. Consider the set
$$ \overline{\Omega} = \{ h:V \to \mathbb{C} \text{: h} \text{ antilinear } \}$$
I am reading Birkenhake, Christina; Lange, H. (1992), Complex Abelian varieties. The books says that the set
$$ \overline{L} = \{ h\in \overline{\Omega} : Im \text{ }h(L) \subset \mathbb{Z} \}$$
Is a lattice for $\overline{\Omega}$ i.e a discrete subgroup of maximal rank. I have no idea how to prove it. The book uses the fact that the bilinear map
$$ \overline{\Omega} \times V \to \mathbb{R}, \text{ } (h,v) \mapsto Im \text{ } h(v) $$
is nondegenerated. I know how to prove the latter fact but not how to use it to prove the proposition. Please help.
