  Motivated by applications to SPDEs we extend the It\^o formula for the square
of the norm of a semimartingale $y(t)$ from Gy\"ongy and Krylov (Stochastics
6(3):153-173, 1982) to the case \begin{equation*} \sum_{i=1}^m \int_{(0,t]}
v_i^{\ast}(s)\,dA(s) + h(t)=:y(t)\in V \quad \text{$dA\times \mathbb{P}$-a.e.},
\end{equation*} where $A$ is an increasing right-continuous adapted process,
$v_i^{\ast}$ is a progressively measurable process with values in $V_i^{\ast}$,
the dual of a Banach space $V_i$, $h$ is a cadlag martingale with values in a
Hilbert space $H$, identified with its dual $H^{\ast}$, and $V:=V_1\cap V_2
\cap \ldots \cap V_m$ is continuously and densely embedded in $H$.
  The formula is proved under the condition that $\|y\|_{V_i}^{p_i}$ and
$\|v_i^\ast\|_{V_i^\ast}^{q_i}$ are almost surely locally integrable with
respect to $dA$ for some conjugate exponents $p_i, q_i$. This condition is
essentially weaker than the one which would arise in application of the results
in Gy\"ongy and Krylov (Stochastics 6(3):153-173, 1982) to the semimartingale
above.
