Find the cardinality of the set of all norms on R^n (hint: show that every norm || || : R n → R is continuous).

Find the cardinality of the set of all norms on R n (hint: show that every norm || || : R n ? R is continuous).

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  • Leave a comment if you need any clarifications.

  • Do you also need the proofs for your other question or just the answer? I would say your offered amount is a bit low.

  • Proofs, how much more do you think i should add?

  • Well, it indeed needs three proofs (one for each case ) and will take about an hour to write. I think the offer should be at least tripled.

  • Okay i'll do that

  • I suggest you to also set a later deadline, if possible. 4 hours is too early for this kind of high level questions.

  • Hey Phillip, i just realized, don't i need a lower bound for the Cardinality for the set of all norms

  • Cool. So you do not need the second part of the argument.

  • im confused. I still need a lower bound for the cardinality for the set of all norms

  • Suppose you have a norm || . || . Then for any real number a, a*|| . || is also a norm. So The cardinality of the set of norms is at least that of the continuum. I hope this makes it clear.

  • See also the attached file.

The answer is accepted.