# Clarification: the continuous linear functional for a vector-valued RKHS

In the well-known scalar-valued RKHS case ($\mathcal{Y}\in\mathbb{R}$), let $\mathcal{X}$ be a set, then the Hilbert space of functions $\mathcal{H}\subseteq\mathcal{Y}^{\mathcal{X}}$ is an RKHS if the linear functional (its evaluation functional) $\delta_{x}:h\mapsto y$ is continuous $\forall x \in \mathcal{X}$.

My question relates to the generalisation $\mathcal{Y}\in\mathbb{R}^{n}$ and the definition of a vector-valued RKHS. Why do we show as in equation (2.1) that we need to verify the linear functional $\delta_{?}:h\mapsto \langle y,h(x) \rangle_{\mathcal{Y}}$ is continuous? More specifically, why is the mapping (that we want to show as continuous on $\mathcal{H}$) maps to an inner product of $y$ and the evaluated function $h(x)$ in the space $\mathcal{Y}$? I know this is a definition but I don't understand the intuition of the evaluation functional mapping to an inner product in the output space.

• What is RKHS? I'm not sure how well known it is... – copper.hat Dec 15 '16 at 16:53
• A Reproducing Kernel Hilbert Space – rnoodle Dec 15 '16 at 16:54