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- How do I go about showing the unit sphere is compact in the ##1##-norm without using the fact that norms on ##\mathbb R^n## are equivalent?

In

Now, in showing that all norms in ##\mathbb R^n## are equivalent, the authors show a given norm ##\|\cdot\|## is equivalent to the ##1##-norm (and then by transitivity, we have equivalence for all norms, since equivalent norms is an equivalence relation). I have already managed to understand that the identity is continuous from ##(\mathbb R^n,\|\cdot\|_1)## to ##(\mathbb R^n,\|\cdot\|)##. To show that the inverse of the identity map is continuous, the authors claim that the unit sphere in the ##1##-norm is compact. I'm getting hung up on this statement, since I don't know how to go about this without using that the norms are equivalent already. How would one show the unit sphere in the ##1##-norm is compact?

I know of Heine-Borel, but I'm not sure how and if it applies here. Any help would be very appreciated.

*Introduction to Topology*by Gamelin and Greene, I'm working an exercise to show the equivalence of norms in ##\mathbb R^n##. This exercise succeeds another exercise where various equivalent formulations of "equivalent norms" have been given, e.g. that two norms ##\|\cdot\|_a,\|\cdot\|_b## are equivalent iff the identity map from ##(\mathbb R^n,\|\cdot\|_a)## to ##(\mathbb R^n,\|\cdot\|_b)## is bicontinuous.Now, in showing that all norms in ##\mathbb R^n## are equivalent, the authors show a given norm ##\|\cdot\|## is equivalent to the ##1##-norm (and then by transitivity, we have equivalence for all norms, since equivalent norms is an equivalence relation). I have already managed to understand that the identity is continuous from ##(\mathbb R^n,\|\cdot\|_1)## to ##(\mathbb R^n,\|\cdot\|)##. To show that the inverse of the identity map is continuous, the authors claim that the unit sphere in the ##1##-norm is compact. I'm getting hung up on this statement, since I don't know how to go about this without using that the norms are equivalent already. How would one show the unit sphere in the ##1##-norm is compact?

I know of Heine-Borel, but I'm not sure how and if it applies here. Any help would be very appreciated.