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Let G be a LCA Topological Group, I define it's dual group, $\Gamma$, by the group of continuous characters $$\gamma:G\to\mathbb{T}$$ where $\mathbb{T}$ is the complex unit circle. I want to equip this group with the Compact-Open topology as defined in Fox (1945) as the topology generated by the sub-basis of sets of function $M(A,W)$ where A is a compact subset of G and W an open set of $\mathbb{T}$. I am struggling slightly with proving that this is a locally compact topology on $\Gamma$ using only the definition I have (that being that a space is locally compact if every element has a compact neighbourhood) and not using any Measure Theory (as is used in Fourier Analysis on Groups by W. Rudin). Is this possible? I know it will suffice to show that $\Gamma$ has a compact neighbourhood of the identity, so I only need to find a compact neighbourhood of the character $$\gamma (x)=1 \forall x\in G$$ but I am struggling slightly to wrap my head around how to do so.

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