= Solution
In the real Hilbert space $L^2(\mathbb R_+)$, let $f_t=1_{[0,t]}$. Then $k_i(t)=\langle f_t,h_i\rangle$. The <Parseval identity for a Hilbertian basis> gives
$$
\begin{aligned}
K(s,t)&=\sum_i\langle f_s,h_i\rangle\langle f_t,h_i\rangle\\
&=\langle f_s,f_t\rangle
=\int_0^\infty1_{[0,s]}(u)1_{[0,t]}(u)\,du
=\min(s,t).
\end{aligned}
$$
The series is absolutely convergent by the <Cauchy-Schwarz inequality>, since $\sum_i k_i(s)^2=s$ and $\sum_i k_i(t)^2=t$. Thus
$$
\boxed{K(s,t)=s\wedge t,}
$$
the <Brownian covariance kernel>. This is the <Brownian covariance from an integrated orthonormal basis>.
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