/sci/ - Science & Math

>>11484595
What? All vectors in a normed linear space, Banach or not, have finite norm; I don't know why you think mentioning non-metrizable things is relevant. Just because a space doesn't have the conventional notion of a norm doesn't automatically mean it has vectors with infinite norm, which was what Remi's was saying.
Most non-metrizable spaces people encounter (in the e.g. weak or Mackey topology) can still be endowed with semi-norms. The problem there isn't with vectors having "infinite norm", it's that they can't be specified by a single norm.
>>11484606
>The norm of an element is the sum of the modulos of it's coefficients in the base.
In general Zorn guarantees a basis in a generic vector space but it doesn't have to be countable. Secondly, you can't really say [math]|v| = \sum_a |\lambda_a|[/math] unless you have a notion of orthonormality, which not even separability guarantees, and this is modulo convergence and completeness issues.
Though it's true that if you want a useful notion of a norm/semi-norm it better be finite.

>>10414547
>students that end an assignment question asking them to [math]verify[/math] Stokes's theorem with "QED"