(More generally, one often encounters “almost circular reasoning” issues in these sorts of problems, in which one almost has to assume the result one is trying to prove as a hypothesis. Often, one has to deal with the apparent circularity by an induction argument, a bootstrap argument, a continuity argument, an infinite descent argument, or passing to the “first counterexample” or “last example”. The above is an instance of the latter strategy.)

]]>In the proof of theorem 14 one can use the continuity of scalar multiplication at (0,0) to make U bounded. This is easier and also shows that the continuity of addition is not needed in this step.

best regards,

pavel

*[Good point; I've made the change. -T.]*