RKHS beginners guide.
Basic Hilbert Space Theory
Normed Vector Space
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 on $\mathcal{F}$ if:
- $\lVert f \rVert_{\mathcal{F}} = 0$ .iff $f = 0$ (norm separates points),
- $\lVert \lambda f \rVert_{\mathcal{F}} = \lvert \lambda \rvert \space \lVert f \rVert_{\mathcal{F}},\space\forall\lambda\in\mathbb{R},\space\forall f \in \mathcal{F}$ (positive homogeneity),
- $\lVert f + g \rVert_{\mathcal{F}} \leq \lVert f \rVert_{\mathcal{F}} + \lVert g \rVert_{\mathcal{F}},\space\forall f, g \in \mathcal{F}$ (triangle inequality).
In every normed vector space, one can de ne a metric induced by the norm: $$ d(f, g)=\lVert f - g \rVert_{\mathcal{F}}. $$
For Example: