I know that a vector space equipped with a seminorm can be transformed into a normed space by taking the equivalence classes of the equivalence relation $f \sim g$ iff $\|f-g\| = 0$.
Can a similar procedure be applied to make a vector space equipped with a positive sesquilinear form into an inner product space? If so, what is the associated equivalence relation one should take? Simply $f \sim g$ iff $\langle f-g,f-g\rangle = 0$?