gn.general topology - Is every norm in R^n a continuous function?

I agree that this looks like a homework question, but since some people have already bitten, I'd just like to point out what might be said for infinite-dimensional spaces. So suppose you have an infinite-dimensional real or complex vector space, equipped with a norm || . ||

What does it mean for a function on V to be continuous? Well, you have to specify a topology on V, and it's natural to use the one defined by the norm. But then it's an immediate corollary of the triangle inequality that the norm function is continuous with respect to the topology it defines. (In some sense, if this weren't true, then we wouldn't bother studying normed vector spaces!)

However, V might also carry some weaker topology (such as a w*-topology induced by some predual) and then the norm will not in general be continuous with respect to that topology.

(Silly remark: equip R^n with the indiscrete topology, i.e. the one with only two members. Then the usual norm is not continuous. Of course, that's a ridiculous topology to put on the space. I have a feeling that every Hausdorff topology on R^n for which translations and dilations are continuous, is equivalent to the usual one, but I'd need to check in something like Rudin's book to be sure.)