Prove that there is $v$ that satisfies specific inner product conditions

Question

I have question from paper, to prove that for the following conditions over $C$

$(i)$ $[\alpha v,u]=\alpha [v,u]$

$(ii)$ $[v,u]=[u,v]$

there is $v\neq0$ for which not satisfies $[v,v]\gt0$

if its not understandable, ill try to correct

Answer 1

So over $\mathbb{C}$ the standard inner product satisfies conditions

• $\langle av,w\rangle = a\langle v,w\rangle$ for all $a \in \mathbb{C}$ (i.e. bilinearity in the first factor)
• $\langle v,w\rangle = \overline{\langle v,w\rangle}$ (i.e. its Hermitian).

This Hermitian property is important, and the fact that you have the condition that $[v,w]=[w,v]$ instead means you should look to exploit this somehow.

To do this consider $[ia,ia]$ for some $a \in \mathbb{C}$. Then we have:

$[ia,ia] \stackrel{\text{prop } i)}{=} i[a,ia] \stackrel{\text{prop } ii)}{=} i[ia,a] \stackrel{\text{prop } i)}{=} i^2[a,a] = -[a,a]$.

So just pick any vector $v \in C$, then by the above we must have either $[v,v] \leq 0$ or $[iv,iv] \leq 0 $.