Let’s call a function a **near-endomorphism** of if there is a constant such that for all . The set of near-endomorphisms of will be denoted by . We put an equivalence relation on by declaring that iff the function is bounded, and let denote the set of equivalence classes.

It’s not difficult to show that defining in terms of pointwise addition and in terms of composition of functions turns into a commutative ring. And it turns out that this ring has a more familiar name… Before reading further, can you guess what it is?

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