 Solution

Use the convention that a sesquilinear form is linear in its first argument and conjugate-linear in its second: for complex scalars $a,b$,No symmetry or positivity is assumed. Put $\chi =\varphi -\psi$. Its diagonal vanishes. Expansion at $v+w$ giveswhile expansion at $v+iw$ givesThese two equations force $\chi (v,w)=\chi (w,v)=0$. This is the polarization argument for a vanishing quadratic form, and proves $\varphi =\psi$ on every pair of vectors.

For the final deduction, the linear map $\alpha$ makes $\psi (v,w)=\varphi (\alpha v,\alpha w)$ another sesquilinear form. Its diagonal agrees with that of $\varphi$, so the result just proved gives