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This article focuses on a new concept of quadratic variation for processestaking values in a Banach space $B$ and a corresponding covariation. This ismore general than the classical one of M\'etivier and Pellaumail. Those notionsare associated with some subspace $\chi$ of the dual of the projective tensorproduct of $B$ with itself. We also introduce the notion of a convolution typeprocess, which is a natural generalization of the It\^o process and the conceptof $\bar \nu_0$-semimartingale, which is a natural extension of the classicalnotion of semimartingale. The framework is the stochastic calculus viaregularization in Banach spaces. Two main applications are mentioned: onerelated to Clark-Ocone formula for finite quadratic variation processes; thesecond one concerns the probabilistic representation of a Hilbert valuedpartial differential equation of Kolmogorov type.

The covariation for Banach space valued processes and applications