Yuan Lu — Notes

Reproducing Kernel Hilbert Space Basics

2026-07-17T00:00:00+00:00

RKHS beginners guide.</p>

Basic Hilbert Space Theory</a></h2>
Normed Vector Space</a></h3>
let $\mathcal{F}$ be a vector space over the field $\mathbb{R}$ of real numbers (or $\mathbb{C}$). A function $\lVert \cdot \rVert_{\mathcal{F}}: \space\rightarrow [0, \infty)$ is said to be a norm</em> on $\mathcal{F}$ if:</p>

$\lVert f \rVert_{\mathcal{F}} = 0$ .iff $f = 0$ (norm separates points),</li>
$\lVert \lambda f \rVert_{\mathcal{F}} = \lvert \lambda \rvert \space \lVert f \rVert_{\mathcal{F}},\space\forall\lambda\in\mathbb{R},\space\forall f \in \mathcal{F}$ (positive homogeneity),</li>
$\lVert f + g \rVert_{\mathcal{F}} \leq \lVert f \rVert_{\mathcal{F}} + \lVert g \rVert_{\mathcal{F}},\space\forall f, g \in \mathcal{F}$ (triangle inequality).</li> </ul>
In every normed vector space, one can dene a metric induced by the norm: $$ d(f, g)=\lVert f - g \rVert_{\mathcal{F}}. $$</p>
For Example:</p>
Inner Product</a></h3>
Definitions</a></h2>
Intuition</a></h2>
Derivation</a></h2>
Examples</a></h2>
Connections</a></h2>
References</a></h2>