To summarize the question I'm going to ask: for those who have studied L-functions and class field theory, I am confused about the definitions of some things and haven't found a good reference for them. So I'm asking about what some things should mean. Let $K$ be a number field, and let $\mathbb A_K$ be the ring of adeles. In my notes on class field theory, I recall a set $C_c^{\infty}(\mathbb A_K)$, the set of $\mathbb{C}$-valued continuous functions on $\mathbb A_K$ which are "smooth" and of "compact support." Being of compact support is straightforward: the closure of the set of $x \in \mathbb A_K$ which are not mapped to $0$ is compact.
"Smooth," however, can mean different things. A function $f: \mathbb{R} \rightarrow \mathbb{C}$ or $f: \mathbb C \rightarrow \mathbb C$ is called "smooth" if it is infinitely differentiable. On the other hand, a function $f: K_v \rightarrow \mathbb{C}$ ($K_v$ being some finite extension of $\mathbb{Q}_p$) would be called smooth if it is locally constant (each point has a neighborhood on which the function is constant).
I am guessing a "smooth" function $f: \mathbb{A}_K \rightarrow \mathbb{C}$ on the adeles would be one of two things:
(i) It is locally constant on the adeles.
or (ii) The restriction of $f$ to each completion $K_v$ is smooth (which means locally constant or infinitely differentiable, depending on whether the place $v$ is finite or infinite)
Another thing: I remember us considering functions of the form $f_0 \otimes f_{\infty}: \mathbb{A}_K \rightarrow \mathbb{C}$, where $f_0$ is a function on the restricted direct product $\prod\limits_{v < \infty} K_v$ and $f_{\infty}$ is a function on $\prod\limits_{v \mid \infty} K_v$. One then defines $f_0 \otimes f_{\infty}(x) := f_0(x) f_{\infty}(x)$. I distinctly recall that one of the things was true:
(i) every $f \in C_c^{\infty}(\mathbb A_K)$ is a finite sum of functions of the form $f_0 \otimes f$.
or (ii) the set of such finite sums is dense in $C_c^{\infty}(\mathbb{A}_K)$.
So my question is: which one of the options (i), (ii) is the correct one?