 Solution

Choose independent random variables ${\xi }_i\sim N(0,1)$ on a countable product probability space. For each fixed $t$, the diagonal assumption givesThe partial sums ${\sum }_{i\le n}{k}_i(t){\xi }_i$ are therefore Cauchy in $L^2$. DefineChoose 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 $t_1,\dots ,t_m$ and real coefficients $a_j$, the linear combination ${\sum }_j{a}_j{X}_{t_j}$ is the $L^2$ limit of centered Gaussian variablesTheir 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 givesThis is the Gaussian process construction from square-summable features.