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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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1 Answer

up vote 7 down vote accepted

The classical proof of this uses a measure and hence integral (Fourier transform) that can be defined because of the topological properties of the underlying group. In case of finite Abelian groups, this boils down to the discrete topology, in case of infinite ones you need the local compactness to ensure what is called the Pontryagin duality. See e.g. http://en.wikipedia.org/wiki/Pontryagin_duality

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Thanks. Your answer is quite helpful. –  Kaveh Apr 29 '11 at 7:48
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