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**Mauritzvdworm** Suppose X is a Hilbert A-module. I would like to show that for each $\displaystyle x\in X$ there is a unique $\displaystyle y\in X$ such that $\displaystyle x=y\cdot\langle y,y\rangle$

The plan is to use adjoinable operators for each $\displaystyle x\in X$

$\displaystyle D_x:X\rightarrow A,D_x(y):=\langle x,y\rangle$

$\displaystyle L_x:A\rightarrow X,L_x(a)=x\cdot a$

where we take $\displaystyle A_A$ to be a right Hilbert A-module with the inner product defined by $\displaystyle \langle a,b\rangle:=a^*b$

I have shown that the operators $\displaystyle D_x,L_x$ are the adjoint of each other