From wikipedia inner product page:
the expected value of product of two random variables is an inner product
$\langle X,Y \rangle = \operatorname{E}(X Y)$. How it can be generalized in case of random vectors?
Or more generally for any probability measure.
Let $\mathbb{P}$ be a set of all probability measures defined on $X$, and let $\mathbb{M}$ be the linear span of $\mathbb{P} – \mathbb{P}$.
How an inner product can be defined on $\mathbb{M} \times \mathbb{M}$?
I've looked to the norm like
$$\|P – Q\|= \sup_{f} \left| \int f \, dP – \int f \, dQ \right|$$
But it seems that this norm doesn't satisfy the parallelogram law (so $\langle x, y\rangle = \frac{1}{4}( \|x + y\|^{2} – \|x – y\|^{2})$ trick cannot be used). Is it possible to proof this?
Best Answer
$\mathbb M$ would be the space of signed measures on $X$ (presumably with respect to a particular $\sigma$-algebra). This is a Banach space with the total-variation norm, but not a Hilbert space, and so it doesn't have a natural inner product.