I'm in a Hilbert space $H$ and for $z,v, h \in H$ and $t \in \mathbb C$ I have
$ \|z\|^2 \leq \|h−(tv+y)\|^2 = \|z−tv\|^2 =\|z\|^2 −2\Re(t⟨v,z⟩)+|t|^2\|v\|^2$
According to my notes it follows from this that $\Re(t⟨v,z⟩) = 0$ for all $t$. How does that follow? I can't seem to show it. Thanks for your help.