Uniformly Equivalent Metrics

  • Thread starter muzak
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Homework Statement


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]|.


Homework 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.

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?
 

Answers and Replies

  • #2
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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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