It is well known that the reproducing kernel hilber space (RKHS) with reproducing kernel K can be characterized as $$ H(K) = \left\{f: f \in L^2[0,T], \|f\|_K^2=\langle f,f\rangle_K<\infty\right\} $$ where $$\langle f,f\rangle_K = \sum_{j=1}^{\infty}\langle f,\phi_j\rangle^2/\lambda_j$$ and where $\lambda_j,\phi_j$ are eigenvalues and eigenfunctions of the kernel $K$.
Let $h \in H(K)$ and $g \in L^2[0,T]$, my question is does $$ \langle h,g\rangle_K = \sum_{j=1}^{\infty}\langle h,\phi_\rangle\langle g,\phi_j\rangle/\lambda_j <\infty $$
My initial guess is that $<h,g>_K$ is bounded. Here are my reasonings:
Since $\sum_{j=1}^{\infty}\langle h,\phi_j\rangle^2/\lambda_j<\infty$, then for each $j$, one must have $\langle h,\phi_j\rangle^2<\infty$, and one could find a lower bound $w$ of the set $\{\langle h,\phi_j\rangle^2>0\}$ such that $$ \sqrt{w} \sum_{j=1}^{\infty} |\langle h,\phi_j\rangle|/\lambda_j < \sum_{j=1}^{\infty}\langle h,\phi_j\rangle^2/\lambda_j<\infty $$ which leads to $$ \sum_{j=1}^{\infty} |\langle h,\phi_j\rangle|/\lambda_j <B<\infty$$ Thus, $$\langle h,g\rangle_K \leq M \sum_{j=1}^{\infty}\langle g,\phi_j\rangle $$ Similarily, since $$ \|g\|_{L^2}^2 = \sum_{j=1}^{\infty} \langle g,\phi_j\rangle^2<\infty$$ one has $$ \sum_{j=1}^{\infty} |\langle g,\phi_j\rangle|<\infty$$ Putting these inequalies,one concludes $$\langle h,g\rangle_K<\infty$$.