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!