Here is my doubt. In case it's not clear what the notation in the title means, I'll report it here:
$\begin{equation*}\mathbb C\left[ X\right]\colon =\left\{ \sum\limits_{x\in X} a_x\cdot e_x\:\middle\lvert\:a_x\in\mathbb C\quad\forall x\in X \right\}\end{equation*}$
Where in $\mathbb C\left[ X\right]$ there is an $e_x$ for each $x\in X$. Specifically I indicate with it the set of formal linear combinations of symbols $e_x$, where two of them are the same one iff they share all the coefficients. We proposed in our class that this space is isomorphic to the set of functions $a\colon X\mapsto \mathbb C$ with compact support.
In my notes I read that there exists an isomorphism between $\mathbb C\left[ X\right]\otimes\mathbb C\left[ Y\right]$ and $\mathbb C\left[ X\times Y\right]$ if $X, Y$ are two non-empty, countable sets, that looks like this:
$\begin{equation*} \mu\left( a\colon X\times Y\mapsto \mathbb C \right)\end{equation*}\quad\colon \quad a\stackrel{\mu}{\longmapsto} \sum\limits_{(x,y)\in X\times Y} a(x,y)\;\delta_x\otimes\delta_y$
Where $\delta_x\colon X\mapsto\mathbb C$ maps $y\in X$ to $1$ iff $x=y$, $0$ otherwise, while $\delta_y\colon Y\mapsto\mathbb C$ maps $x\in X$ to $1$ iff $y=x$, $0$ otherwise.
Now, my uneducated mathematical mind has the following doubt: shouldn't $\delta_x,\delta_y$ be something in $\mathbb C\left[X\right]$ and $\mathbb C\left[Y\right]$ respectively? Like $\delta_x$ in my intuition should map $y\in X$ to $e_x$ iff $x=y$, $\:0$ otherwise and same thing for $\delta_y$. Am I tripping?
EDIT: Sorry, forgot to mention that in case $X$ is an infinite set, only a finite amount of coefficients $a_x$ is allowed to be non-zero.
The $\delta_x$ are implicitly of the form you want them. Consider a formal symbol of the form $\sum a_x \cdot e_x$. We could think of this as a function $\mathbb{X} \to \mathbb{C}$ by considering $\sum a_x \cdot \delta_x$.
Conversely, a function $f: X \to \mathbb{C}$ that is non-zero at finitely many values corresponds to the element $\sum f(x) \cdot e_x$.