Let a vector space V be given. Let f:V→V have the property that for all x,y,a∈V,
f(x+a)−f(y+a)=f(x)−f(y)
Q1. I'd like to know how weak one can make additional assumptions to guarantee that f is affine, namely that there exists a linear transformation L:V→V, and an element v∈V such that
f(x)=L(x)+v
for all x∈V.
I think, for example, that if one assumes that V=Rn, and if f is differentiable, then (⋆) implies that f is affine. This makes me think that perhaps one needs a notion of smoothness or continuity in general and therefore possibly a norm? Also
Q2. Is there a common name for the property (⋆)?
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