The proof of the Paley-Zygmund inequality provided in the link just says that the Cauchy-Schwarz inequality was used but doesn't show the steps. I tried to read and understand the wiki article on Cauchy-Schwarz to derive Paley-Zygmund inequality but couldn't understand the below line:
After defining an inner product on the set of random variables using the expectation of their product, $X \cdot Y = E(XY)$
Can you please help me understand the meaning of inner product on the set of random variables? Any directions on how to derive the Paley-Zygmund inequality would also be helpful.