Hermitian form, inner product and symplectic form relationship

See before: complexification of a vector space.

(See at Calibre library the document: Complexification, complex structures, inner products, symplectic forms and linear differential equations, page 4)

To sum up, in a real vector space $V$ with a complex structure $J$, one real inner product determine a symplectic form and viceversa.

Moreover, an inner product $h$ defined in $V$ seen as a complex vector space (keep an eye: an Hermitian inner product) determine in $V$ an inner product $g$ (in the real sense) and a symplectic form $\omega$ in such a way that

If B is a bilinear form on $V$ then we say that $J$ preserves B if

for all $u,v\in V$.
An equivalent characterization is that $J$ is skew-adjoint with respect to B:

If $g$ is an inner product on $V$ then $J$ preserves $g$ if and only if $J$ is an orthogonal transformation. Likewise, $J$ preserves a nondegenerate, skew-symmetric form $\omega$ if and only if J is a symplectic transformation (that is, if $\omega (Ju,Jv)=\omega (u,v)$). For symplectic forms $\omega$ there is usually an added restriction for compatibility between $J$ and $\omega$, namely

for all non-zero $u\in V$. If this condition is satisfied then $J$ is said to tame $\omega .$
Given a symplectic form $\omega$ and a linear complex structure $J$, one may define an associated symmetric bilinear form $g_J$ on $V_J$