Reproducing Kernel Hilbert Space (RKHS) constructed by the summation of positive-definite kernels.

Question

Let $k_1,\ldots,k_p$ be positive definite kernels defined on $\mathcal{X}\times\mathcal{X}$, where $\mathcal{X}$ is a non-empty set. $k_i$ is the reproducing kernel of the Reproducing Kernel Hilbert Space (RKHS) $\mathcal{H}_i$, which is endowed with an inner product $\langle\cdot,\cdot\rangle_{\mathcal{H}_i}$, from which the norm $\lVert\cdot\rVert_{\mathcal{H}_i}$ is induced ($i=1,\ldots,p$).

Now let $k$ be a real-valued function defined on $\mathcal{X}\times\mathcal{X}$, with $k=\sum_{i=1}^{p}k_i$. It is straightforward to prove (isn't it?) that $k$ is positive definite, since positive-definiteness is preserved under summation. Apparently, there corresponds a RKHS, $\mathcal{H}$ to $k$, with inner product $\langle\cdot,\cdot\rangle_{\mathcal{H}}$ and norm $\lVert\cdot\rVert_{\mathcal{H}}$.

What is the relation between the norm of $\mathcal{H}$ and the norms of $\mathcal{H}_i$, $i=1,\ldots,p$? Do we need to require that the spaces $\mathcal{H}_i$ are pairwise-perpendicular so that it holds

$$ \lVert\cdot\rVert_{\mathcal{H}}^2 = \sum_{i=1}^{p}\lVert\cdot\rVert_{\mathcal{H}_i}^2, $$

and, if so, what that would mean for the reproducing kernels $k_i$?

Moreover: Let $f\in\mathcal{H}$, how could its norm, $\lVert f \rVert_{\mathcal{H}}$, be expressed in terms of the norms of the spaces $\mathcal{H}_i$?

The above questions may seem vague, or even incorrect in the way they are stated. Please feel free to suggest approaches and/or corrections (for instance, should the title be changed?), if necessary. Also, please feel free to discuss!

Answer 1

If $k_1,\dots,k_p$ are kernels with RKHSs $\mathcal{H}_i$, then $k=\sum_i k_i$ is indeed a positive definite kernel again without further assumptions. The relation $\lVert f\rVert_{\mathcal{H}}^2 = \sum_{i=1}^{p}\lVert f\rVert_{\mathcal{H}_i}^2$ holds automatically for all functions that are in the intersection of all participating RKHSs. There is no additional condition of "pairwise-perpendicularity".

The interesting question is, which functions are in this intersection? Because of the RKHS construction and the dominance of norms one might guess $\mathcal{H}\subseteq \mathcal{H_i}$, so the intersection is $\mathcal{H}$ itself. But that would still require a proof.