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Uniformly Equivalent Metrics

  1. Sep 10, 2011 #1
    1. The problem statement, all variables and given/known data
    Prove that in R[itex]^{n}[/itex], the euclidean metric, the d[itex]_{\infty}[/itex]=max{|a1-b1|,...,|a[itex]_{n}[/itex]}-b[itex]_{n}[/itex]|}, and d = |a1-b1|+...|a[itex]_{n}[/itex]}-b[itex]_{n}[/itex]|.


    2. Relevant equations
    Uniform Equivalence: basically p,d so that we have the two inequalities with some constants like p(x,y)[itex]\leq[/itex]Ad(x,y) and d(x.y)[itex]\leq[/itex]Bp(x,y).
    Schwarz inequality.

    3. The attempt at a solution
    I was going to do this in straightforward manner but when we go to see what our constants are, they turn out to n or [itex]\infty[/itex]. I don't know what to do. Can we treat them as coefficients in the two respective ineqaulities?
     
  2. jcsd
  3. Sep 10, 2011 #2

    micromass

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    Well, n is a good constant, but [itex]\infty[/itex] is not. Where did you get [itex]\infty[/itex]?? Then we'll look if we can fix that.
     
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