I am reading the book "Brownian Motion, Martingales, and Stochastic Calculus", Jean-François Le Gall and I have a doubt regarding the Gaussian vector definition.
First, the enveiroment is $(\Omega, \mathcal{F}, \mathbb{P})$ a proability space and $E$ a $d$-dimensional Euclidean space with inner product. When he defines a Gaussian vector, he states that a "radom variable" $X$ is a Gaussian vector with values in $E$ if $\forall\, u \in E$, $<u, X>$ is a (real) Gaussian variable.
But what is the definition of the inner product $<u,X>$?
As far as I know, the inner product is only defined for elements of $E$: $$<.,.> E \times E \to \mathbb{R}$$
How to correctly define the inner product between an element of $E$ and a "random variable" $X: \Omega \to E$?