Understanding the definition of Hilbert function space

Question

I was reading the Hilbert function spaces from the chapter 2 of Pick Interpolation and Hilbert Function Spaces by Jim Agler & John E. McCarthy. It says the following-

So by Hilbert function space we mean the collection of functions from a set $X$ to $H^*$ for some Hilbert space $H$ where $H^*$ denotes the dual of $H$. Is it correct? Or, I'm misinterpreting the definition?

But this is the definition the Bergman space (in Example 2.2 above) is becoming a Hilbert function space?

Can anybody help me to understand the definition of Hilbert function space? Thanks for your help in advance.

Answer 1

No, $\cal H$ is the function space, not the codomain of the functions, that is not precised in the definition.

More formally, $\cal H$ is an Hilbert function space if $\cal H$ is a Hilbert space and $$ {\cal H} = \{f:X\to Y \,|\, \forall x\in X, (x\mapsto f(x)) \text{ is continuous} \} $$ for some sets $X$ sand $Y$.

In the case of the example, ${\cal H} = L^2_a(\Bbb D)$, $X = \Bbb D$ and $Y = \Bbb C$.