- #1

Dafe

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## Homework Statement

Let [tex]||\cdot |[/tex]| denote any norm on [tex]\mathbb{C}^m[/tex]. The corresponding dual norm [tex]||\cdot ||'[/tex] is defined by the formula [tex]||x||^=sup_{||y||=1}|y^*x|[/tex].

Prove that [tex]||\cdot ||'[/tex] is a norm.

## Homework Equations

I think the Hölder inequality is relevant: [tex]|x^*y|\leq ||x||_p ||y||_q, 1/p+1/q=1[/tex] with [tex]1\leq p, q\leq\infty[/tex]

## The Attempt at a Solution

Since a norm is a function satisfying three properties, I need to show that they hold.

(1) [tex]||x||'=0[/tex] if and only if [tex]x=0[/tex].

(2) [tex]||\alpha x||'=|\alpha| ||x||^[/tex].

(3) [tex]||x+z||'\leq ||x||^+||z||^[/tex].

I manage to do (1) and (2) just fine, but the triangle inequality (3) is giving me problems.

I use the Hölder inequality to get the following:

[tex]||x+z||'=sup_{||y||=1}|y^*(x+z)|\leq ||y|| ||x+z||=||x+z||[/tex]

[tex]||x||'=sup_{||y||=1}|y^*x|\leq ||y|| ||x|| =||x||[/tex]

[tex]||z||'=sup_{||y||=1}|y^*z|\leq ||y|| ||z|| =||z||[/tex]

(1) [tex]||x||'\leq ||x||[/tex]

(2) [tex]||z||'\leq ||z||[/tex]

(3) [tex]||x||'+||z||' \leq ||x||+||z||[/tex]

I also know that

(4) [tex]||x+z|| \leq ||x||+||z||[/tex]

I am unable to show that [tex]||x+z||\leq ||x||'+||z||'[/tex] which I think I must if I am to prove the triangle inequality.

Any help is appreciated.