Every symplectic space has complex structure?

Question

Let $(E,\omega)$ be a symplectic space, Is what can always be provided by a complex structure ?

In the following cases

i) E is of infinite dimension;

ii) E is a real vector space;

iii) E is a complex vector space;

Thank you in advance

Answer 1

If $E$ is a vector space take a scalar product $b$, and define $J(x)$ by $\omega(x,y)=b(J(x),y)$.

If $(E,\omega)$ is a manifold, such a complex structure implies that $(M,\omega, J)$ is Kahler and this is not always true.

https://mathoverflow.net/questions/107795/examples-of-non-kahler-compact-symplectic-manifolds