Two notions of equivalent norms?

by Gory   Last Updated June 22, 2018 17:20 PM

I have two statements in mind that taken without further caution could seem contradictory:

  • all norms are equivalent in finite dimension
  • there are infinitely many non-equivalent norm over the rationals (Ostrowski)

So I should be missing a point. Is it that the first statement is only valid for the reals or complexes? (however I have the impression that the proof still holds over the rationals)

Or is it rather than the two notions of equivalence (one with bounds, continuity of the identity; the other with equality up to a certain power) are different? Case in which: why are these two natural, what motivates one in some cases and the other one in others?

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