Wednesday, 31 December 2014

linear algebra - What extra assumption makes this transformation affine?

Let a vector space V be given. Let f:VV have the property that for all x,y,aV,
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:VV, and an element vV such that
f(x)=L(x)+v

for all xV.



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 ()?

No comments:

Post a Comment

real analysis - How to find limhrightarrow0fracsin(ha)h

How to find limh0sin(ha)h without lhopital rule? I know when I use lhopital I easy get $$ \lim_{h\rightarrow 0}...