Banach space valued random variable Let $X$ be a Banach space valued random variable. 
Is there a characteristic function of $X$ in this case? How it is defined if there is one?Are there any applications of this function in this high dimensional case? For example proving limit theorems?
What if $X$ takes values in a (locally convex) topological vector space?
 A: As far as I know, there are two definitions for Banach values random variables:
1.  The Pettis integral: $E(X) = Y$ means $E(\xi(X)) = \xi(Y)$ for any $\xi$ in the dual space;
2.  The Bochner integral: $X$ is a limit of simple functions, and is defined just as the regular real valued integral.
I think most books and papers go with the Bochner integral.  This is equivalent to $X$ being measurable, and taking values in a separable subspace except on a set of measure zero.
A characteristic function is obviously defined in a Pettis integral like manner, and I think this is the only way.
An interesting example of a non-Bochner inegrable function: Let $[0,1] \to M([0,1])$, where $M$ denotes the space of signed Radon measures (ie the dual of $C([0,1])$, given by the map $x\mapsto \delta_x$.
A: As you suspect, a characteristic function or Fourier transform for a random variable $X$ valued in a Banach space $B$ is usually defined as the function $\phi_X : B^* \to \mathbb{C}$ defined by $\phi_X(f) = \mathbb{E} e^{i f(X)}$.  Here $B^*$ is the continuous dual of $B$.
For example, we say $X$ is Gaussian iff $f(X)$ is Gaussian for all $f \in B^*$, iff $\phi_X(f) = e^{-q(f)/2}$ where $q$ is a quadratic form on $B^*$.
