(Anti-)hermitian structure on dual space Let $H$ be an (anti-)hermitian form on a complex vector space $V$. 1)Does $H$ induce a natural (anti-)hermitian form on the dual $V^*$? 2)Suppose $f$ is unitary with respect to $H$. Is there any natural way to make the transpose $f^T$ unitary with respect to something induced by $H$?

1) If $H$ is a Hermitian form on $V$, it induces an isomorphism $V\to V^*$ by $v\mapsto(u\mapsto H(u,v))$. Now that $V^*$ is identified in a specific way with $V$, it can have the same Hermitian form.

2) Let $f:V\to V$ be unitary, and let $\varphi_v\in V^*$ be given by $\varphi_v(u)=H(u,v)$. By the above identification, we have $\|\varphi_v\|=\|v\|$. By definition of the transpose operator, we have $$f^t(\varphi_v)(u)=\varphi_v(f(u))=H(f(u),v)=H(u,f^*(v)),$$ thus $f^t(\varphi_v)$ is the functional identified with $f^*(v)$, and $\|f^t(\varphi_v)\|=\|f^*(v)\|$. Since $f$ is unitary, so is $f^*$, and finally $$\|f^t(\varphi_v)\|=\|f^*(v)\|=\|v\|=\|\varphi_v\|,$$ hence $f^t$ is unitary with respect to the induced Hermitian form.