> [!definition] > > Let $X$ be a [[Set|set]] and $H \subset \complex^X$ be a [[Hilbert Space|Hilbert space]] consisting of functions $X \to \complex$. A mapping $K: X^2 \to \complex$ is a **reproducing kernel** if > 1. For each $x \in X$, $K(\cdot, x) \in H$. > 2. For every $f \in H$, $f(x) = \angles{f, K(\cdot, x)}$. > > $H$ admits a reproducing kernel if and only if $\delta_x: H \to \complex$ defined by $f \mapsto f(x)$ is [[Bounded Linear Map|bounded]].