Let $f:V\times V \rightarrow \Bbb C$ be a bilinear form in a finite inner product space. Will there always be a single linear transformation $T:V\rightarrow V$ for which $f(v,u) = \langle Tv,u\rangle$ for each $v,u\in V$. If not - what's an example of this not happening?
(I can prove this for bilinear forms $f:V\times V \rightarrow \Bbb R$, but can't see if the same applies to $\Bbb C$).