Proving that the characters of an infinite Abelian group is a basis for the space of functions from the group to $\mathbb{C}$

Let $A$ be an arbitrary (possibly infinite) Abelian group. A character $\chi$ is a group homomorphism from $A$ to the multiplicative group of complex numbers.

I can prove that if $A$ is finite then the characters form a basis for the set of functions from $A$ to $\mathbb{C}$, but how can we prove this for an infinite Abelian group? I guess that we might need to add extra assumptions like the group being locally compact, but I am not sure.

So here is my question:

How can we prove that the set of characters forms a basis for the space of functions from $A$ to $\mathbb{C}$? Do we need any extra assumptions on $A$?

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