# Variance of function-valued random variables

I am struggling with abstract definitions of basic statistical concepts.

For a random variable $X$ which we assume to live in a real Hilbert (or even Banach) space $\mathcal{H}$, its expectation is defined as an element $\mathbb{E}[X] \in \mathcal{H}$ such that

$$\mathbb{E}[y^*(X)] = y^*(\mathbb{E}[X])$$ for all $y^{*} \in \mathcal{H}^{*}$, the dual space of $\mathcal{H}$. This is fine and to simplify, let's assume that $\mathbb{E}[X] = 0$.

How can I then define the variance of $X$? Is it correct to say that it's then equal to $\text{Var}(X) = \mathbb{E}[(y^{*}(X))^2]$. Won't it depend on the choice of $y^{*}$?

I am pretty confused here. Please let me know if the statements aren't clear (or true).

EDIT: any textbook references on this kind of theory will be kindly appreciated!

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Does this help? – martini May 3 '12 at 16:16
Where did you get this definition of expectation for Hilbert spaces? – Manuel Schmidt Oct 29 '14 at 8:16
Bosq "Theory of linear processes in function spaces". – johnny Nov 12 '14 at 11:37