Weight-preserving isomorphisms between spaces of continuous functions: The scalar case

Let FF be a finite field (or discrete) and let AA and BB be vector spaces of FF-valued continuous functions defined on locally compact spaces X and Y  , respectively. We look at the representation of linear bijections H:A⟶BH:A⟶B by continuous functions h:Y⟶Xh:Y⟶X as weighted composition operators. In order to do it, we extend the notion of Hamming metric to infinite spaces. Our main result establishes that under some mild conditions, every Hamming isometry can be represented as a weighted composition operator. Connections to coding theory are also highlighted.