  Functional It\^o calculus was introduced in order to expand a functional
$F(t, X\_{\cdot+t}, X\_t)$ depending on time $t$, past and present values of
the process $X$. Another possibility to expand $F(t, X\_{\cdot+t}, X\_t)$
consists in considering the path $X\_{\cdot+t}=\{X\_{x+t},\,x\in[-T,0]\}$ as an
element of the Banach space of continuous functions on $C([-T,0])$ and to use
Banach space stochastic calculus. The aim of this paper is threefold. 1) To
reformulate functional It\^o calculus, separating time and past, making use of
the regularization procedures which matches more naturally the notion of
horizontal derivative which is one of the tools of that calculus. 2) To exploit
this reformulation in order to discuss the (not obvious) relation between the
functional and the Banach space approaches. 3) To study existence and
uniqueness of smooth solutions to path-dependent partial differential equations
which naturally arise in the study of functional It\^o calculus. More
precisely, we study a path-dependent equation of Kolmogorov type which is
related to the window process of the solution to an It\^o stochastic
differential equation with path-dependent coefficients. We also study a
semilinear version of that equation.
