# $L_2$ norm and RHKS norm in Hilbert spaces $\mathcal{H}$

According to this paper (just right below the Theorem 3 and above the section 3)

Reproducing Kernel Hilbert Space(RKHS) $\mathcal{H}$ on $\mathcal{X}$ is a Hilbert space of functions from $\mathcal{X}$ to $\mathbb{R}$. $\mathcal{H}$ is an RKHS if and only if there exists a $k(x,x'):\mathcal{X}\times\mathcal{X}\to\mathbb{R}$ such that $\forall x\in\mathcal{X},\,k(x,\cdot)\in\mathcal{H}$, and $\forall f\in\mathcal{H},\, \langle f,k(x,\cdot)\rangle_{\mathcal{H}}=f(x)$. If such a $k$ exist, it is unique and it is a PD kernel. A function $f\in\mathcal{H}$ if and only if $\|f\|^2_{\mathcal{H}}=\langle f,f\rangle_{\mathcal{H}}<\infty$, and its $L_2$ norm is dominated by RKHS norm $\|f\|_{L_2}\le \|f\|_{\mathcal{H}}$.

How the last claim can be done ?

I thought this would be fundamentally mathematical, and the claim and related definitions are independent of this specific paper, thus it might be understood especially by people who had some background knowledge. I want to know how to prove this claim, and I am a beginner and even do not know how to describe $\|f\|_{L_2} \leq \|f\|_{\mathcal{H}}$ explicitly .

• Begin by stating the definition of $\mathcal H$. – user147263 Dec 26 '14 at 3:30
• I don't really understand what you're asking. By definition, a Hilbert space is a (complete) inner product space. So if $f\in\mathcal H$, then necessarily $\langle f,f\rangle$ is a complex number (with imaginary part zero by conjugate symmetry but that isn't important here); there is nothing to prove there. As for the second part, I imagine it would depend on which measure space with respect to which you're taking the $L^2$ norm... – Math1000 Dec 26 '14 at 3:31
Use the definition with $f\colon x'\mapsto k(x,x')$: we get $$k(x,x) =\lVert k(x,\cdot)\rVert_{\mathcal H} ^2.$$ By Cauchy-Schwarz inequality, we get $$|f(x)|^2\leqslant|k(x,x)|^2\lVert f\rVert_\mathcal H^2,$$ hence $\lVert f\rVert_{L^2}\leqslant \lVert f\rVert_\mathcal H$ provided that $\int_{\mathcal X }|k(x,x)|^2\leqslant 1$.