I am stuck with an exercise that I found in a textbook by Conway. First, I would like to clarify what is meant by a semi-inner product.
Definition. Suppose that $\mathscr X$ is a vector space over the complex field $\mathbb C$. A semi-inner product on $\mathscr X$ is a function $u:\mathscr X\times\mathscr X\to\mathbb C$ such that for all $\alpha,\beta$ in $\mathbb C$, and $x,y,z$ in $\mathscr X$, the following are satisfied:
- $u(\alpha x+\beta y,z)=\alpha u(x,z)+\beta u(y,z)$,
- $u(x,x)\ge 0$,
- $u(x,y)=\overline{u(y,x)}$,
where $\bar\alpha$ is the complex conjugate of $\alpha$.
The difference between an inner product and a semi-inner product is that an inner product also satisfies the following:
- if $u(x,x)=0$, then $x=0$.
Now I formulate the exercise from the textbook.
Let $u(\cdot,\cdot)$ be a semi-inner product on $\mathscr X$. Then $\left|u(x,y)\right|^2=u(x,x)u(y,y)$ if and only if there are $\alpha$ and $\beta$ in $\mathbb C$, not both $0$, such that $u(\beta x+\alpha y,\beta x+\alpha y)=0$.
How can I show that if there are $\alpha$ and $\beta$ in $\mathbb C$, both not $0$, such that $u(\beta x+\alpha y,\beta x+\alpha y)=0$, then $\left|u(x,y)\right|^2=u(x,x)u(y,y)$?