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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 tothe usual It\^o-integral for c\`adl\`ag-integrands. The goal of this note is tocomplete this result and to provide the full connection to the It\^o-integral.We also sketch an application to stochastic partial differential equations.

A note on stochastic integrals as $L^2$-curves