Suppose $\mathcal{H}$ is a reproducing kernel Hilbert space with reproducing kernel $K$. Assume that $f,g$ are two elements in $L_{2}$ and also in $\mathcal{H}$.
My question is what kind of condition will guarantee that $$ \|f\|_{\mathcal{H}} \leq \|g\|_{\mathcal{H}} \iff \|f\|_{L_{2}} \leq \|g\|_{L_{2}}, $$ where $\|\cdot\|_{\mathcal{H}}$ is the norm of RKHS, or is there anyway that I can directly prove it?