The problem involves finding positive coefficients \(a\) and \(b\) such that \(a||x||_{\infty} \leq ||x|| \leq b||x||_{\infty}\). The context centers around the equivalence of norms in a vector space.
The discussion is ongoing, with participants seeking clarification on definitions and attempting to establish a foundational understanding of the norms involved. Some have suggested exploring specific cases, such as \(n=2\), to aid in visualizing the problem.
There is a noted requirement for participants to demonstrate effort in their attempts, as the original poster has not yet provided any initial work or attempts at the problem.
lep11 said:Homework Statement
Find coefficients a,b>0 such that a||x||∞≤||x||≤b||x||∞.Homework Equations
The Attempt at a Solution
No idea how to get started. Help will be appreciated.
I have no idea how to find the coefficients...and I am not expecting you to do my homework.Math_QED said:You must have tried something? You must have thought something? Share it with us because you are required to show some effort.
lep11 said:I have no idea how to find the coefficients...and I am not expecting you to do my homework.
it has something to do with equivalence of norms?
lep11 said:Homework Statement
Find coefficients a,b>0 such that a||x||∞≤||x||≤b||x||∞.Homework Equations
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@lep11, definitions of these two norms would have been useful in the (empty) Relevant equations section.pasmith said:How is ∥x∥ defined?
How is ∥x∥∞ defined?
pasmith said:How is [itex]\|x\|[/itex] defined?
[itex]\|x\|_{\infty}[/itex]:=max{|x1|,...,|xn|}pasmith said:How is [itex]\|x\|_{\infty}[/itex] defined?
x∈Rn, so dimension of the space is n.pasmith said:What is the dimension of the space on which these norms are defined?
lep11 said:##\|x\|:=(\sum_{i=1}^{n} x_i^2)^½##
[itex]\|x\|_{\infty}[/itex]:=max{|x1|,...,|xn|}x∈Rn, so dimension of the space is n.