Conditional Expectation as Inner Product

Question

Hansen and Richard (1987), Page 592, introduces a conditional counterpart to an inner product and also defines a conditional norm. They define for random variables X and Y and $\sigma$-field G $<X,Y>_G = E(XY|G)$ and $\|X\|_G=\sqrt{<X,X>_G}$. Since $E(XY|G)$ is a G-measurable Random Variable and not a Real or a Complex Number, can this be interpreted as an inner product in an extended sense? Are there any references for its use in Probability Literature? I have a similar question about the conditional norm too.