A particular step in the existence proof of the conditional expectation of a sq integrable rv

[Quasar Chunawala]

Hey guys,

One of the standard results in elementary stochastic calculus is that, if $X$ is a random variable, and $Y$ is any square integrable random variable, then the conditional expectation $\mathbb{E}[Y|X]$ exists and is unique. I am struggling to understand a particular step (inequality) in the first part of the proof, where the author proves that $W=g(X)$ and $(Y-Y^{\star})$ are orthogonal. I have posted this as a question on MSE here.

I have tried searching the proof online, but the approach in my text appears a bit different. I would really benefit from any suggestions/clues with it.

Cheers,
Quasar

 

[jahaanrawat]

There is a theorem about existance and uniqueness of orthogonal projections.

 

[Quasar Chunawala]

Hey, I was struggling to understand a particular step in the existence proof; it's all clear now. I posted it as an answer to my question on MSE! Thanks.