Hilbert spaces induced by Hilbert space valued functions

Abstract:

Let $E$ be an arbitrary set and $\mathcal {F}(E)$ a linear space composed of all complex valued functions on $E$. Let $\mathcal {H}$ be a (possibly finite-dimensional) Hilbert space with inner product ${(,)_\mathcal {H}}$. Let ${\mathbf {h}}:E \to \mathcal {H}$ be a function and consider the linear mapping $L$ from $\mathcal {H}$ into $\mathcal {F}(E)$ defined by ${({\mathbf {F}},{\mathbf {h}}(p))_\mathcal {H}}$. We let $\tilde {\mathcal {H}}$ denote the range of $L$. Then we assert that $\tilde {\mathcal {H}}$ becomes a Hilbert space with a reproducing kernel composed of functions on $E$, and, moreover, it is uniquely determined by the mapping $L$, in a sense. Furthermore, we investigate several fundamental properties for the mapping $L$ and its inverse.