Equivalence of Metrics in R^{n}

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In summary, the problem is to prove that in R^n, for the euclidean metric, d_{\infty}=max{|a1-b1|,...,|a_{n}}-b_{n}|} and d = |a1-b1|+...|a_{n}}-b_{n}| are uniform equivalent. The attempt at a solution involves using the Schwarz inequality, but the issue arises when trying to find appropriate constants for the inequalities. While n can be used as a constant, \infty is not suitable. Further discussion is needed to address this issue.
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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?
 
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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.
 

FAQ: Equivalence of Metrics in R^{n}

What is a uniformly equivalent metric?

A uniformly equivalent metric is a type of mathematical function that measures the distance between two points in a space. It is defined as a metric that is topologically equivalent to another metric, meaning that the two metrics produce the same open sets and thus the same topology.

How is uniform equivalence different from other types of equivalence?

Uniform equivalence differs from other types of equivalence, such as topological equivalence or homeomorphic equivalence, in that it focuses specifically on the distance function between points rather than the overall structure of the space. This makes it a more precise measure of equivalence for metric spaces.

What is the significance of uniformly equivalent metrics?

Uniformly equivalent metrics are important in mathematics because they allow us to compare different metrics on the same space and determine if they are topologically equivalent. This can help us understand the structure and properties of a space, as well as provide a basis for developing new metrics.

Can uniformly equivalent metrics be used interchangeably?

No, uniformly equivalent metrics cannot be used interchangeably. While they may produce the same open sets and topology, they may differ in other important properties, such as convergence or completeness. Therefore, it is important to carefully consider which metric is most appropriate for a given problem or situation.

How do we prove that two metrics are uniformly equivalent?

To prove that two metrics are uniformly equivalent, we must show that they satisfy certain conditions, such as being Lipschitz equivalent or having the same convergence properties. This typically involves demonstrating that the two metrics produce the same open sets and topology, as well as showing that one metric can be transformed into the other through a series of continuous functions.

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