= Solution
For the construction above, independence of the <normal random variables> makes $S_N(h)$ normal with mean zero and variance $v_N=\sum_{j\le N}\langle h,e_j\rangle^2$. Its <characteristic function> is $\exp(-\theta^2v_N/2)$. The $L^2$ convergence gives $L^1$ convergence, and
$$
\left|\mathbb E e^{i\theta S_N(h)}-\mathbb E e^{i\theta X(h)}\right|\le |\theta|\,\mathbb E|S_N(h)-X(h)|\longrightarrow0.
$$
Since $v_N\to\|h\|^2$ by the <Parseval identity for a Hilbertian basis>, the limiting <characteristic function> identifies
$$
\boxed{X(h)\sim N(0,\|h\|^2).}
$$
This includes $h=0$, where the <normal distribution> is degenerate at zero.
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