Let $H$ be an (uncountable) set of homomorphisms from an additive group $\Gamma$ to $\mathbb{C}$. Is it possible to define a linear structure on $H$ (over $\mathbb{R}$ or $\mathbb{C}$)? In case it is not always possible how can one prove that given $H$ cannot become linear no matter how the actions of addition and multiplication by scalars are defined?
2026-02-23 04:40:38.1771821638
Existence of a linear structure in a space of homomorphisms
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