Extending $\widetilde{V}$ in order to obtain a partial isometry operator

Question

Recall the following definition:

I want to get a detailed proof of the following theorem. In particular the importance of extending $\widetilde{V}$.

Answer 1

1. A linear map $U:A\to B$ is called an isometry here whenever $\|Ua\|=\|a\|$ for all $a\in A$. Note that surjectivity is not imposed, so that $\tilde V:\mathrm{ran}(P)\to H$ is an isometry, and by continuity, so is its extension to the closure of $\mathrm{ran}(P)$.
   An important point to note is that $\tilde V$ is a well defined map, because whenever $Px=0$, we have $Tx=0$.

2. Since $H=\overline{\mathrm{ran}(P)} \oplus \mathrm{ran}(P)^\perp$, we can define $V(x+y)$ as $\lim_n\tilde V(x_n)$ for a sequence $x_n\to x$, where $x_n\in\mathrm{ran}(P)$ and $y\perp\mathrm{ran}(P)$.
   Now its kernel is the second component, and it is an isometry restricted to the first component.