Let $(\Omega, \mathcal A, Prob)$ be a probability space. Let $V$ be a set of all random $n$-vectors $X$ of the form $X = [x_1 \cdots x_n]^T$, where each $x_i: \Omega \rightarrow \mathbb R$ is in turn an arbitrary random variable with finite second moment, i.e., $e\left[X^T X\right] < \infty$. Is $V$ a real inner product space without requiring that expectation $E[X] = 0$ for each element $X$ of $V$? If yes, what is the simplest inner product definition that does the job?
(Note: $E$ is the expectation operator on $V$ whereas $e$ is the expectation operator on the vector space of all random variables $x: \Omega \rightarrow \mathbb R$.)
It is clear to me that $V$ satisfies the axioms of a real vector space. I can also see that if $V$ were to consist of only those random $n$-vectors $X$ that have zero expectation, then the inner product definition $\langle X, Y \rangle = e\left[X^TY\right]$ transforms $V$ into a real inner product space. However I am unsure about the general case of $V$ as mentioned in the question. Would appreciate some insight, and a simple example of an inner product definition, if the answer to the question is "yes". Thanks.