Let $V$ be a complex vector space and an anti-linear involution $J:V \rightarrow V$ (this means that $J^2 = I$ and if $\lambda \in \mathbb{C}$ and $x, y \in V$ we have $J(\lambda x + y) = \overline{\lambda} J(x)+J(y)$). Fix $N$ a natural number and consider a multilinear functional $\omega:V^{2N}\rightarrow \mathbb{C}$ such that:
- $\omega(x_1,...,x_{2N}) = \omega(x_2, ..., x_{2N},x_1)$, for every $x_1,x_2,...,x_{2N} \in V$
- The matrix $A$ whose matrix elements $A_{ij}$ defined by $A_{ij} = \omega(Jx_{i1},..., Jx_{iN}, x_{j1},..., x_{jN})$ is positive semi-definite for every $x_{kl} \in V$, $k=1,..., n$ and $l=1,...,N$, where $n$ can be any natural number.
I saw this on a paper that I'm currently reading and the author claims that the condition 2 implies that the following: $$\langle x, y \rangle = \omega(x_1,..., x_N, Jy_N,... Jy_1)$$
defines an inner product in $V^N$. I'm not sure about this because this is not linear in the first coordinate. Did I get this wrong?