  Let $X:=(X_1, \ldots, X_p)$ be random objects (the inputs), defined on some
probability space $(\Omega,{\mathcal{F}}, \mathbb P)$ and valued in some
measurable space $E=E_1\times\ldots \times E_p$. Further, let $Y:=Y = f(X_1,
\ldots, X_p)$ be the output. Here, $f$ is a measurable function from $E$ to
some Hilbert space $\mathbb{H}$ ($\mathbb{H}$ could be either of finite or
infinite dimension). In this work, we give a natural generalization of the
Sobol indices (that are classically defined when $Y\in\mathbb R$ ), when the
output belongs to $\mathbb{H}$. These indices have very nice properties. First,
they are invariant. under isometry and scaling. Further they can be, as in
dimension $1$, easily estimated by using the so-called Pick and Freeze method.
We investigate the asymptotic behaviour of such estimation scheme.
