Sesquilinear form/inner product

Question

I have just a quick question: what is the relationship/difference between a sesquilinear form $F(u,v)$ and a complex inner product $(u,v)$?

Answer 1

A complex inner product $F$ is by definition a sesquilinear form, with some additional properties: we should have $F(v,v)\geq 0$ for each $v$ and $F(v,v)=0$ if and only if $v=0$.

You can construct in a vector space of dimension $2$ example of sesquilinear form such that we have $F(v,v)=0$ for all $v$ but $F(v_0,v_0)=0$ for some $v_0\neq 0$.

But a sesquilinear form doesn't need to be an inner product, for example $F(u,v)=0$ is obviously sesquilinear but not an inner product since we can have $F(v,v)= 0$ if $v\neq 0$ (except in the trivial case in which the vector space consists only of ${0}$).