I want to be precise in some notation I am using and could use a verification/review. I have some function $f(\mathbf{A},\mathbf{b})=\mathbf{A}\mathbf{b}$ where $\mathbf{A} \in \mathbb R_{[0,1]}^{J\times K}$ and $\mathbf{b}\in\mathbb R_{[0,\infty]}^K$.
I use this to mean $\mathbf{A}$ is a matrix with dimensions $J \times K$ and its elements are continuous on the interval $[0,1]$ and then $\mathbf{b}$ is a vector of length $K$ with positive elements, including $0$. I don't think for $\mathbf{b}$ I should use $\mathbb R^+$ because that would not include $0$.
This gives a typical matrix result such that their product gives a vector of length $J \times 1$.
The mapping I would use in this case I think would be $f \colon \mathbb R_{[0,\infty]} \to \mathbb R_+^J$.
In plain language I think this means the function with inputs on the domain $[0,\infty]$ maps to the set of positive real numbers giving a vector of length $J$.
It is the mapping where I am most uncertain and could use a critique/review/suggestion, although all comments are helpful.
If we let $\newcommand{\A}{\mathbf{A}} \newcommand{\b}{\mathbf{b}} \newcommand{\RJK}{\mathbb{R}_{[0,1]}^{J \times K}} \newcommand{\RK}{\mathbb{R}_{[0,\infty]}^K} f(\A,\b) := \A\b$, then in words $f$ is a function which takes in a matrix $\A \in \RJK$ and a vector $\b \in \RK$ and sends them to the vector $\A\b$.
Thus, $f$ has domain $\RJK \times \RK$ and codomain $\mathbb{R}^J_{[0,\infty]}$, if anything, i.e. $$ f : \RJK \times \RK \to \mathbb{R}^J_{[0,\infty]} $$ would be the correct notation. There might be a technical sense in which you think of $f$ as taking in $JK+K$ positive real numbers (but certainly not $1$, as your notation suggests), and sending them to some vector, but the way you defined the function we are looking at matrix and some vector being sent to their product.
In general, if a function $g$ sends the pair $(s,t)$ (where $s \in S$ and $t \in T$) to some evaluation of the function $g(s,t)$, we would say that $g$ has domain $S \times T$, where $S \times T = \{(s,t) \mid s \in S, T \in T\}$.