Thread: 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

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