= Solution
For a finite list of real coefficients $(a_j)_{j\leq m}$,
$$
\sum_{j=1}^m a_j\xi_j=\int_0^1\left(\sum_{j=1}^ma_jg_j(t)\right)\,dW_t.
$$
A deterministic <Itô integral> is centered Gaussian: first verify it for step functions as a linear combination of independent Gaussian Brownian increments, then pass to an L2 approximation using the <Itô isometry> and characteristic functions. Its <variance> is the squared L2 norm of the integrand, which is $\sum_ja_j^2$ by the supplied orthonormality. Therefore
$$
\mathbb E\exp\left(i\sum_{j=1}^ma_j\xi_j\right)=\exp\left(-\frac12\sum_{j=1}^ma_j^2\right)
=\prod_{j=1}^m e^{-a_j^2/2}.
$$
This factors as the joint <characteristic function> of independent standard normals. Since every finite subfamily has this law, \b[the entire sequence is independent and each $\xi_n$ has law N(0,1)]. This is the <Gaussian coordinates of deterministic orthonormal Wiener integrands> principle.
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