Gaussianity of deterministic Brownian stochastic integrals (${\int }_0^{t}h(s)d{B}_s\sim N(0,{\int }_0^{t}h(s{)}^{2}ds)$)

For deterministic square-integrable $h$, approximate by step functions. Their Itô integrals are sums of independent normal Brownian increments. The Itô isometry gives convergence in $L^2$, and the variances converge to $\int h^2$; convergence of characteristic functions proves the normal limit. Each linear combination of finitely many such integrals is another deterministic stochastic integral, so they are jointly Gaussian.