= Sum of reproducing-kernel Hilbert spaces
{title2=$\mathcal H_{\sum k_j}$}
For <positive-semidefinite kernels> $k_j$ with <Reproducing-kernel Hilbert spaces> $\mathcal H_j$, the <Reproducing-kernel Hilbert space> of $k=\sum_jk_j$ consists of sums $f=\sum_jf_j$, with
$$
\|f\|_{\mathcal H_k}^2=\min_{\sum_jf_j=f}\sum_j\|f_j\|_{\mathcal H_j}^2.
$$
The minimum is attained uniquely. Indeed, in the <Hilbert space> direct sum $E=\bigoplus_j\mathcal H_j$, the tuples summing to zero form a closed subspace $N$: they are the intersection, over inputs $x$, of the kernels of the continuous maps $(f_j)\mapsto\sum_jf_j(x)$. Every affine fibre has a unique representative in $N^\perp$ by <orthogonal decomposition by a closed subspace>. The <vector> $(k_j(\cdot,x))_j$ lies in $N^\perp$ and represents evaluation in the quotient; its evaluation at $y$ is $\sum_jk_j(y,x)$, identifying the <positive-semidefinite kernel> with $k$.
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