  This paper discusses a new notion of quadratic variation and covariation for
Banach space valued processes (not necessarily semimartingales) and related
It\^o formula. If $\X$ and $\Y$ take respectively values in Banach spaces
$B_{1}$ and $B_{2}$ and $\chi$ is a suitable subspace of the dual of the
projective tensor product of $B_{1}$ and $B_{2}$ (denoted by
$(B_{1}\hat{\otimes}_{\pi}B_{2})^{\ast}$), we define the so-called
$\chi$-covariation of $\X$ and $\Y$. If $\X=\Y$, the $\chi$-covariation is
called $\chi$-quadratic variation. The notion of $\chi$-quadratic variation is
a natural generalization of the one introduced by M\'etivier-Pellaumail and
Dinculeanu which is too restrictive for many applications. In particular, if
$\chi$ is the whole space $(B_{1}\hat{\otimes}_{\pi}B_{1})^{\ast}$ then the
$\chi$-quadratic variation coincides with the quadratic variation of a
$B_{1}$-valued semimartingale. We evaluate the $\chi$-covariation of various
processes for several examples of $\chi$ with a particular attention to the
case $B_{1}=B_{2}=C([-\tau,0])$ for some $\tau>0$ and $\X$ and $\Y$ being
\textit{window processes}. If $X$ is a real valued process, we call window
process associated with $X$ the $C([-\tau,0])$-valued process $\X:=X(\cdot)$
defined by $X_t(y) = X_{t+y}$, where $y \in [-\tau,0]$. The It\^o formula
introduced here is an important instrument to establish a representation result
of Clark-Ocone type for a class of path dependent random variables of type
$h=H(X_{T}(\cdot))$, $H:C([-T,0])\longrightarrow\R$ for not-necessarily
semimartingales $X$ with finite quadratic variation. This representation will
be linked to a function $u:[0,T]\times C([-T,0])\longrightarrow \mathbb{R}$
solving an infinite dimensional partial differential equation.
