How can we prove that a continuous additive homomorphism $ \Phi \colon \mathbb{R}^{n}\to \mathbb{R}^{m} $ is $\mathbb{R}$-linear. i.e. satisfies $ \Phi (rv)=r \Phi (v)$ for $ r\in \mathbb{R} $ and $ v \in \mathbb{R}^{n} $?
proof that a continuous additive homomorphism $\mathbb{R}^n\to\mathbb{R}^m$ is $\mathbb{R}$-linear
1
$\begingroup$
linear-algebra
abstract-algebra
-
0they are not homework problems – 2012-03-04
-
1Then, when you post them, please say (i) in what context you are encountering these problems; and (ii) what your thoughts on the problems so far are. When one sees the same person posting multiple homework-level problems with little or no work showing (in three of the four you provided no work), it sure *looks* like homework. – 2012-03-04
-
0What can you say about the set of those $r\in\mathbb R$ which satisfy $\Phi(rv)=r\Phi(v)$ for all $v\in\mathbb R^n$? – 2012-03-19
-
7I think it is pretty absurd to offer a bounty for «credible and/or official sources» when there are two *proofs* offered! If what is wanted is more details, then more details should be asked for, but asking for «credible and/or official sources» is very weird :) – 2012-03-19
-
1To make the question more fun, drop the continuity hypothesis and replace it by Lebesgue measurability. – 2012-03-22
-
1@Mariano: I don't know about yours, but my degree says I have "all rights and privileges pertaining thereto"; I'd say we are both "official sources", to say nothing of our credibility! – 2012-03-23