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I've been trying to use the cauchy-schwarz inequality to prove that $(cov(X,Y))^2 \leq var(x)var(Y)$. The cauchy-schwarz inequality can be expressed as follows: if $u$ and $v$ are vectors in an inner product space then $^2 \leq (\parallel u\parallel)^2 (\parallel v\parallel)^2$. How do you define the vectors and the inner product so to prove the result in the first sentence. I've seen something like $(cov(X,Y))^2 = < (x-E(X)),(y - E(Y)) > \leq (\parallel (x-E(X) \parallel )^2(\parallel (y-E(Y) \parallel )^2 = var(X)var(Y) $ but how is $< (x-E(X)),(y - E(Y)) >$ expressed as a sum? Also how can you let $(x-E(X))$ and $(y - E(Y))$ be vectors?

Any insight would be great.

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