Hi,


I've been wondering about the following :


Is it possible, without the axiom of choice, to have two inequivalent complete norms on a vector space?


All the examples of inequivalent complete norms I've seen rely on the existence of Hamel bases...


This is most likely well-known, but I'd be glad if someone could provide a good reference.


Thank you,
Malik



Inequivalent complete norms and the axiom of choice
No, it is not possible. There is a model due to Shelah of ZF+ dependent choice+every set of reals has the Baire property. There is a result of Garnir and Wright that implies that in such model, any two complete norms are equivalent. The [Handbook of Analysis and Its Foundations](http://www.math.vanderbilt.edu/~schectex/ccc/index.html) by Eric Schechter has a chapter on this result and its consequences. The chapter is "The Dream Universe of Garnir and Wright". The result in that chapter is supposed to be slightly stronger than the one in the original papers, with which I'm not familiar.


