It's well known that the characteristic function of a gaussian measure $\mu$ on $\mathbb{R}^d$ is given by: $$ \hat{\mu}(\xi)=\int_{\mathbb{R}^d}exp(i\xi . x)\mu (dx),\qquad \xi \in \mathbb{R}^d $$ but when we move to an infinite dimensional space; let's say a Banach space $X$ the formula becomes: $$ \hat{\mu}(f)=\int_{X}exp(if(x))\mu (dx),\qquad f\in X^* $$ My question is how such formula has been derived ? what analysis leads to this standing on the finite dimensional result ?
Thank you for your time.