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Let $\mathcal{B}(F)$ the algebra of all bounded linear operators on a 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.

I see in a paper that $\langle\cdot,\cdot\rangle_M$ induces an inner product on the quotient space $F/\text{Ker}(M)$. I don't understand how to construct this inner product?

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Check yoursef that $$\langle[x],[y]\rangle = \langle Mx,y\rangle$$ ($[\cdot] =$ class of equivalence) is well-defined.

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