Let $\mathcal{B}(F)$ the algebra of all bounded linear operators on an **infinite dimensionel** complex Hilbert space $(F,\langle\cdot,\cdot\rangle)$. Let $M\in \mathcal{B}(F)^+$ (i.e. $\langle Mx\;, \;x\rangle\geq 0$ for all $x\in F$). Then
$$\langle\cdot,\cdot\rangle_{M}:F\times F\longrightarrow\mathbb{C},\;(x,y)\longmapsto\langle Mx, y\rangle,$$
is a semi-inner product.

$\langle\cdot,\cdot\rangle_M$ induces an inner product on the quotient space $F/\text{Ker}(M)$ defined as:
$$\langle[x],[y]\rangle = \langle Mx,y\rangle$$
($[\cdot]$ denotes the class of equivalence).

I see in [this paper] the following result and the authors refer the reader to ([1]) but I don't find a proof in this reference.

> **Theorem:** The completion of $F/\text{Ker}(M)$ under the above inner product is isometrically isomorphic to the Hilbert space $\text{Im}(M^{1/2})$
with the inner product
$$(M^{1/2}x,M^{1/2}y)_{\text{Im}(M^{1/2})}:=\langle Px, Py\rangle,\;\forall\, x,y \in F,$$
where $P$ denotes the orthogonal projection of $F$ onto the closure of $\text{Im}(M)$.

> I hope to get a sketch of proof of the above theorem.



  [1]: https://www.math.purdue.edu/~branges/square-summable.pdf
  [this paper]: http://dx.doi.org/10.1080/03081087.2012.667094