I showed as an exercise that for an additive function $T: V\to W$, if $T(u+v)= T(u)+T(v)$, then if the field is the rationals, we get that the linearity implies scalar multiplication, i.e. $T(qv)=qT(v)$ for all q in rationals and v in V and so T is actually a linear transformation. My question is, if the field was all of R, can we conclude the same thing? I think not, but is there a way to extend this result by maybe introducing limiting procedures into the vector space? Thanks!
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If $T$ is continuous, then yes: it must be a linear transformation. See Cauchy's functional equation. Also, I'm pretty sure this has been asked on this site before. – Ben Grossmann Feb 07 '17 at 19:54
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See also here. – Feb 07 '17 at 20:03