I'm slightly confused about what $\nabla$ means, and especially what the dot product means, in the context of the definition of $\text{div} \mathbf{F}$.
For example, you will see Folland in his Advanced Calculus textbook define $\text{div} \mathbf{F} = \nabla \cdot \mathbf{F} = \partial_1F_1 + \ldots + \partial_n F_n$.
Since $\mathbf{F}:\mathbb{R}^n \rightarrow \mathbb{R}^n$ is a vector field, it's clear we can write $\mathbf{F} = (F_1, \ldots, F_n)$. This is an $n$ dimensional vector where each component $F_i$ is from the vector space of all real valued functions whose domain is $\mathbb{R}^n$. This part makes sense to me.
The gradient for a function $f:\mathbb{R}^n \rightarrow \mathbb{R}$ is defined as $\nabla f = (\partial_1 f, \ldots, \partial_n f)$. Where I am lost is, what $\nabla$ means without the $f$.
Question 1: If it means $(\partial_1 , \ldots, \partial_n )$, then what space is each element $\partial_i$ from? Furthermore, what is the meaning of $\partial_i$ if it is not next to a $f$?
My Guess: My guess is that $\partial_i : (\mathbb{R}^n)^* \rightarrow (\mathbb{R}^n)^*$, since it takes a real-valued function in $\mathbb{R}^n$ and spits out another one (namely its partial derivative).
Question2: But if my guess is correct in Question 1, how does one take a dot product of $\nabla$ and $\mathbf{F}$ if they're not from the same space? ($F_i$ is from the vector space of all real valued functions whose domain is $\mathbb{R}^n$, where as $\partial_i$ is not.)