How does the product of sets of complex numbers give a character? I'm working through this "Introduction to Banach Algebras" and just after proposition 8.2 they say:

If $A$ is a commutative Banach algebra, $a\in A$ and $\phi\in M(A)$, then $\phi(a)\in sp(a)$. Hence, $\phi\in\prod_{a\in A}sp(a)$

In this context $\phi$ is a character (which they define as a homomorphism between an algebra and $\mathbb{C}$). $M(A)$ is the set of all C-ideals of A; because every C-ideal maps bijectively to a character with that C-ideal as its kernel, they use $M(A)$ ambiguously to refer to characters or C-ideals. 
It makes sense that $\phi(a)\in sp(a)$, but I can't figure out what they mean with the product symbol.

The notation $\prod_{i\in I}S_i$ denotes a set of functions. By definition, $f\in\prod_{i\in I}S_i$ if (i) $f$ is a function with domain $I$ and (ii) $f(i)\in S_i$ for every $i\in I$.
So $\phi\in\prod_{a\in A}sp(a)$. Because $\phi$ is a function with domain $A$ and $\phi(a)\in sp(a)$ for every $a\in A$.

Come to think of it, that raises an obvious objection that should probably be addressed. Ojection:
"What? $$\prod_{i\in\{0,1\}}A_i=\prod_{i=0}^1A_i=A_0\times A_1.$$That's not a space of functions!"
Well ok, maybe $A_0\times A_1$ is not a space of functions. But it's trivially canonically equivalent to a space of functions. Map the ordered pair $(a_0,a_1)$ to the function $f$ such that $f(0)=a_0$ and $f(1)=a_1$.
