Norm inequality, find coefficients

[lep11]

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.

 

[member 587159]

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.

You must have tried something? You must have thought something? Share it with us because you are required to show some effort.

 

[lep11]

You must have tried something? You must have thought something? Share it with us because you are required to show some effort.

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?

 

[pasmith]

How is \|x\| defined?
How is \|x\|_{\infty} defined?
What is the dimension of the space on which these norms are defined?

 

[Ray Vickson]

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?

Yes, that is exactly what the problem is about---showing how to prove norm equivalence.

 

[micromass]

What space is $x$ a member of? How are the norms defined?

 

[Mark44]

Homework Statement

Find coefficients a,b>0 such that a||x||_∞≤||x||≤b||x||_∞.

Homework Equations

<empty>

How is ∥x∥ defined?
How is ∥x∥_∞ defined?

@lep11, definitions of these two norms would have been useful in the (empty) Relevant equations section.

 

[lep11]

How is \|x\| defined?

$\|x\|:=(\sum_{i=1}^{n} x_i^2)^½$

How is \|x\|_{\infty} defined?

\|x\|_{\infty}:=max{|x₁|,...,|x_n|}

What is the dimension of the space on which these norms are defined?

x∈Rⁿ, so dimension of the space is n.

 

[Ray Vickson]

$\|x\|:=(\sum_{i=1}^{n} x_i^2)^½$

\|x\|_{\infty}:=max{|x₁|,...,|x_n|}x∈Rⁿ, so dimension of the space is n.

Why don't you look first at the case of $n=2$, where you can easily draw pictures and make $(x_1,x_2)$-diagrams to help you focus your thinking?

 

[Mark44]

At 9 posts into this thread, the OP still has not shown an attempt -- thread closed.

@lep11, you may start another thread on this question, but you MUST show some effort or there will be consequences.