Do I have to show this map is well-defined?

Question

Let $H$ be a Hilbert space and $u \in B(H)$. Write

$$ H = \overline{\mathrm{im}(u)} \oplus \overline{\mathrm{im}(u)}^\bot$$

and define $v(h) = v(|u|x \oplus z):= u(x)$.

Do I have to prove that $v$ is well-defined? And if so, why?

Answer 1

You have to prove that $|u|x_1\oplus z_1=|u|x_2\oplus z_2$ implies $u(x_1)=u(x_2).$