# Inner product from orthogonality relations

Let us say we want to define an inner product on random vectors in $\mathbb R^n$. Say, $X$ and $Y$ are two such vectors. A natural inner product is $E \langle X, Y \rangle_{\mathbb R^n}$ where the inner product here is the usual Euclidean inner product.

Assume on the other hand, that we take a clue from independence: $X$ and $Y$ are independent iff any (measurable) function of $X$ is uncorrelated from any (meas.) function of $Y$: $E [f(X) g(Y)] = E f(X) E g(Y)$. Let us relax this condition and assume that any linear function of $X$ is uncorrelated from any linear function of $Y$. For simplicity assume that we work with zero-mean random vectors. Thus, we want to say that $X$ and $Y$ are orthogonal'' iff $$E [\langle X, u\rangle \langle Y, v\rangle ] = 0, \quad \forall u,v \in \mathbb R^n. \quad (*)$$ Is there an inner-product defined on the zero-mean random vectors in $\mathbb R^n$ (with finite second moments) that has precisely these orthogonality relations, i.e., $X$ and $Y$ are orthogonal according to it iff (*)? If not, do we have some inner product that comes "close", say the implication goes one way or another?

(*) is equivalent to $E[X \otimes Y] = 0$ or $E(X_i Y_j) = 0$ for all $i,j=1,\dots,n$.