  In a work of van Gaans (2005a) stochastic integrals are regarded as
$L^2$-curves. In Filipovi\'{c} and Tappe (2008) we have shown the connection to
the usual It\^o-integral for c\`adl\`ag-integrands. The goal of this note is to
complete this result and to provide the full connection to the It\^o-integral.
We also sketch an application to stochastic partial differential equations.
