= Solution
Choose independent random variables $\xi_i\sim N(0,1)$ on a countable product probability space. For each fixed $t$, the diagonal assumption gives
$$
\sum_i k_i(t)^2=K(t,t)<\infty.
$$
The partial sums $\sum_{i\leq n}k_i(t)\xi_i$ are therefore Cauchy in $L^2$. Define
$$
\boxed{X_t=L^2\text{-}\lim_{n\to\infty}\sum_{i=1}^n k_i(t)\xi_i.}
$$
Choose a representative of this limit for each $t$. No path continuity or simultaneous series convergence over all uncountably many times is being asserted.
For any finite list $t_1,\ldots,t_m$ and real coefficients $a_j$, the linear combination $\sum_ja_jX_{t_j}$ is the $L^2$ limit of centered Gaussian variables
$$
\sum_{i=1}^n\left(\sum_{j=1}^m a_jk_i(t_j)\right)\xi_i.
$$
Their variances converge, so their <characteristic functions> converge to that of a centered <normal distribution>. This proves that every finite-dimensional vector is Gaussian and hence that $X$ is a <Gaussian process>. Taking $L^2$ limits also gives
$$
\boxed{\mathbb EX_t=0,\qquad
\mathbb E[X_sX_t]=\sum_i k_i(s)k_i(t)=K(s,t).}
$$
This is the <Gaussian process construction from square-summable features>.
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