Euclidean Norm and Maximum Norm

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• Oct 6th 2009, 08:01 PM
6DOM
Euclidean Norm and Maximum Norm
Q: Explain why the Euclidean norm and maximum norm (any two norms on Rn for that matter) result in the same open sets. It follows from this that a sequence Xk in Rn will converge with respect to the maximum norm if and only if it will converge with respect to the Euclidean norm.

I have absolutely no idea how to approach this problem.
Any suggestions?

Thanks as always
• Oct 7th 2009, 04:26 AM
tonio
Quote:

Originally Posted by 6DOM
Q: Explain why the Euclidean norm and maximum norm (any two norms on Rn for that matter) result in the same open sets. It follows from this that a sequence Xk in Rn will converge with respect to the maximum norm if and only if it will converge with respect to the Euclidean norm.

I have absolutely no idea how to approach this problem.
Any suggestions?

Thanks as always

Let A(w,r) be the open ball with center w and radius r wrt the euclidean norm. Prove this set is open also wrt the max. norm (take any point X=(x_1,..,x_n) in A(w,r) and supose w = (w_1,...,w_n) ==> SUM(x_i - w_i)^2 < r^2 ==> but we also have that SUM(x_i - w_i)^2 <= n*max(x_i - w_i)^2, so if we choose wisely and s.t. |y_i - w_i| < r/Sqrt(n) then...etc. and try to end the argument by yourself. The other direction is simmilar.

Tonio