Is every norm in R^n a continuous function? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-26T08:08:01Z http://mathoverflow.net/feeds/question/1282 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/1282/is-every-norm-in-rn-a-continuous-function Is every norm in R^n a continuous function? mike 2009-10-19T20:54:20Z 2009-10-27T04:05:39Z <p>Is every norm in R^n a continuous function?</p> http://mathoverflow.net/questions/1282/is-every-norm-in-rn-a-continuous-function/1284#1284 Answer by John D. Cook for Is every norm in R^n a continuous function? John D. Cook 2009-10-19T20:57:59Z 2009-10-19T20:57:59Z <p>Yes, because in finite dimensional spaces all norms are topologically equivalent.</p> http://mathoverflow.net/questions/1282/is-every-norm-in-rn-a-continuous-function/1286#1286 Answer by Akhil Mathew for Is every norm in R^n a continuous function? Akhil Mathew 2009-10-19T21:10:10Z 2009-10-19T21:10:10Z <p>Yes, and this is true more generally (same reason as John Cook) for norms over a finite-dimensional vector space over a field complete with respect to an absolute value. It doesn't work for infinite-dimensional spaces.</p> http://mathoverflow.net/questions/1282/is-every-norm-in-rn-a-continuous-function/1289#1289 Answer by Yemon Choi for Is every norm in R^n a continuous function? Yemon Choi 2009-10-19T21:28:04Z 2009-10-19T21:28:04Z <p>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 || . ||</p> <p>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!)</p> <p>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.</p> <p>(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.)</p> http://mathoverflow.net/questions/1282/is-every-norm-in-rn-a-continuous-function/2761#2761 Answer by Orr Shalit for Is every norm in R^n a continuous function? Orr Shalit 2009-10-27T04:05:39Z 2009-10-27T04:05:39Z <p>The answer is "yes". </p> <p>The exaplanation depends on what is meant by "continuous". Let's agree that "the norm || || is continuous" means "if (x_n) is a sequence of vectors such that the coordinates x_n converge to the coordinates of some vector x, then ||x_n|| converges to ||x||".</p> <p>Let e_1, ..., e_n be the standard basis of R^n. First one shows that for every i, the function t -> ||t*e_i|| is continuous (t real). Then, after writing every vector as a_1 * e_1 + ... + a_n * e_n, the result follows easily from the triangle inequality. </p>