Suppose we have 2 vector random variables $X, Y\in \mathbb V$, $X: \Omega \rightarrow \mathbb V$ and $Y: \Omega \rightarrow \mathbb V$where $\mathbb V$ is a vector space with inner product $(X,Y)$. I have heard that the inner product of the two random vectors $X, Y$ is defined by $E(X,Y)$ which is scalar.
To define an inner product we must first have a vector space. The random variables $X,Y$ are functions, so does that mean the set of random variables $\{X: X:\Omega \rightarrow \mathbb V\}$ forms a vector space? Could someone please explain this to me.
Thanks a lot.
PS. The question Inner product and norms for random vectors is related but not exactly what I want to know.
Best Answer
The set of functions $\Omega \mapsto \Bbb{V}$ does in fact form a vector space in a natural way through componentwise addition and scalar multiplication.
More formally, given any two elements $X, Y \in \hom(\Omega, \Bbb{V})$, and a scalar $a\in \Bbb{R}$, we define $[X+Y](\omega) = X(\omega) + Y(\omega)$ and $[aX](\omega) = a \cdot X(\omega) $for any $\omega \in \Omega$, using the addition and scalar multiplication in $\Bbb{V}$