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If a map $f : \mathbb{R} \to \mathbb{R}$ is additive, in the sense that $f(x + y) = f(x) + f(y)$, then it is simple to show that $f$ is $\mathbb{Q}$-linear, buy it does not need to be $\mathbb{R}$-linear in general. This is a classical problem, known as Cauchy's functional equation. It turns out that as soon as the function is even remotely regular (continuous, boundend on an interval, or measurable), on can show that it has to be linear, and thus of the form $f(x) = cx$ for some $c \in \mathbb{R}$.

It turns out that after some work, one can prove a similar claim about maps of the torus $T = \mathbb{R}/\mathbb{Z}$: if $f :\ T \to T$ obeys $f(x+y) = f(x) + f(y)$ and is measurable, then it is of the form $f(x) = cx$, where $c \in \mathbb{Z}$.

I think that it follows that additive maps on $T^m$, or $\mathbb{R}^n$, or even $\mathbb{R}^n \times T^m$, are necessarily "linear". I believe it can be shown by applying the $1$-dimensional result to $t \mapsto \pi f(tv)$, where $\pi$ are projections and $v$ are vectors. (For instance, for $T^2$, write $f(t_1,t_2) = (f_{11}(t_{1}) + f_{12}(t_2), f_{21}(t_{1}) + f_{22}(t_2))$ and reason for each $f_{ij}$ independently).

I would like ask two things, and I would be very grateful for either.

  1. Is the result for the torus and/or multidimensional case generally known, and if so what would be a good keyword for further search or a reference? Is there an easy proof for the torus? (my reasoning for the torus goes very much like the standard one for $\mathbb{R}$ I know of, but perhaps there is a clever reduction from one to the other?)

  2. If I am not mistaken about the multidimensional case, it would follow that a an additive map of a commutative Lie group into itself is automatically continuous, and even has a particularly nice form. Is it true that an additive map of a general Lie group into itself is automatically continuous?

Edit/Answer It turns out that the related issues have been studied with much success and are relatively well understood (at least insofar as there are strong results on the topic). The phenomenon in question is known as Automatic Continuity, and has been studied by the Polish mathematical school, and Andre Weil. There are theorems that assure continuity of Baire measurable homomorphisms between Polish group (Banach) and Haar measurable homomorphism from a locally compact Polish group into a Polish group (Weil). I encourage anyone interested to consult this excellent question/answer pair on MathOverflow.

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up vote 0 down vote accepted

As alluded in the edit to the question, an answer can be found on MathOverflow here.

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