Suppose $(V,\omega_V)$ and $(W,\omega_W)$ are two finite-dimensional real symplectic vector spaces. I want to know how I can define new symplectic structures.
I know $V\oplus W$ comes with a natural symplectic form $\omega_V \oplus \omega_W$. How can I define symplectic structures on tensor products and dual spaces?
I have been looking around and haven't found anything regarding this issue.