Let $V$ be a vector space of finite dimension on $\mathbb C$, $B$ a basis of $V$ , and $f, g$ two sesquilinear forms. If $B$ is an orthonormal basis for $f$ and for $g$ then $f=g$.
I want to know if the following statement is true or false. It does not seem false to me because I feel like any vector can be written as a linear combination of elements in $B$ and the images of the basis are equal. But I also feel like the fact we are in $\mathbb C$ gives a counter example I am not thinking of. What do you think ?