 Solution

Approximate $h$ in $L^2[0,t]$ by deterministic step functions $h_n$. Each integral $\int h_{n}dB$ is a linear combination of independent Gaussian increments and hence is Gaussian, with mean zero and variance $\int h_n^2$. The Itô isometry gives convergence in $L^2$ to $\int hdB$, so characteristic functions pass to the limit. Thus