A step in a question I am working on goes as follows:
For some arbitrary Inner product space and two arbitrary elements $x,y$, we have the two results:
Let $a \in \mathbb R$, then:
$2 a \text{ Re} \langle x, y \rangle+ a^2 \| y \| \ge 0 \implies \text{ Re} \langle x, y \rangle = 0$
and similarly if $a \in \mathbb C$, so that $a = bi$ then:
$2 b i \text{ Im} \langle x, y \rangle+ b^2 \| y \| \ge 0 \implies \text{ Re} \langle x, y \rangle = 0$
I don't understand why either of these implications holds obviously
The first term depends on $x$, and you can make it as large and positive (or large and negative) as you like by rescaling $x$, whereas the second term doesn't depend on $x$. Therefore the only way the inequalities can hold is if the first term is zero. It's the same as saying that if $Ax+B \geq 0$ for all real $x$, then $A=0$.