symplectic form on $V$. Then the set $\jj_\omega$ of $\omega$-tame linear complex structures on $V$ is a non-empty open contractible subset in the set $\jj$ of all linear complex structures on $V$*]{}.

Proof. [*Step 1*]{}. The linear version of the Darboux theorem states that there exists a basis $(\bfx_1, \ldots, \bfx_n, \bfy_1, \ldots, \bfy_n)$ of $V$ such that for corresponding linear coordinates $(x_1, \ldots, x_n, 
y_1, \ldots, y_n)$ on $V$ $$%\textstyle
\omega=\sum_{i=1}^n dx_i \wedge dy_i.
\eqno\rom(1.2.3)$$ Set $J_0 (\bfx_i) \deff \bfy_i$, $J_0 (\bfy_i) \deff -\bfx_i$. It is easy to see that $J_0$ is a linear complex structure on $V$. Moreover, the $\cc$-valued linear functionals $z_i \deff x_i + \isl\, y_i$ induce the isomorphism $\phi
:(V,J_0) \cong (\cc^n, J\st)$, $J\st \deff \isl$, of [*complex*]{} linear spaces, such that the given symplectic form $\omega$ on $V$ is mapped into the standard sympletic form $\omega\st$ on $\cc^n$ given by the same formula (A.T.1), $\phi^*\omega\st = \omega$. Thus we can conclude that there exists a $J_0$-Hermitian metric $h$ on $V$ with the following properties:

$g(v,w) \deff \re h(v,w)$ is a symmetric positively definite form on $V$ such that $h(v,w)= g(v,w) + \isl\,\omega(v,w)$ for any $v,w \in V$;

$g(J_0 v, J_0w) = g(v,w)$ and $\omega(J_0 