= Solution
For any $r_1,\ldots,r_n\in\mathbb R$, the linearity from part (a) gives
$$
\sum_{j=1}^n r_jX(h_j)=X\left(\sum_{j=1}^n r_jh_j\right)
$$
almost surely, and the right side has a <normal distribution>. This is the defining linear-combination criterion for a <multivariate normal distribution>; singular <covariance matrices> are allowed.
Passing to the limit in the inner products of the partial sums gives
$$
\boxed{\operatorname{Cov}(X(g),X(h))=\mathbb E[X(g)X(h)]=\sum_j\langle g,e_j\rangle\langle h,e_j\rangle=\langle g,h\rangle.}
$$
The passage to the limit is justified by <Cauchy-Schwarz inequality> and $L^2$ convergence. Equivalently, the vector's <characteristic function> is
$$
\mathbb E\exp\left(i\sum_jr_jX(h_j)\right)=\exp\left(-\frac12\sum_{j,k}r_jr_k\langle h_j,h_k\rangle\right).
$$
Thus both joint normality and the complete <covariance matrix> follow from the Hilbert-space inner product.
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