Finding the dual norm on X' when ||x||=max|ξᵢ|

[azdang]

Homework Statement

X is the space of ordered n-tuples of real numbers and ||x||=max|$$\xi$$_j| where x=($$\xi$$₁,...,$$\xi$$_n). What is the corresponding norm on the dual space X'?

Homework Equations

The Attempt at a Solution

I think the answer is that ||x*||=|x_1|+...+|x_n| , but I'm not sure if that's correct or how to show it. Any ideas? Thanks so much.

 

[matt grime]

What is the definition of the dual space and its norm?

 

[azdang]

Well, I know the dual space,X', is the set of all bounded linear functionals on X and the norm on that space is:
||f||=sup|f(x)|/||x|| for x in X and x not equal to 0
or
||f||=sup|f(x)| for x in X and ||x||=1

 

[azdang]

Hey guys, although Matt advised me to think about the definitions, I'm still confused how to apply them to this problem. Any ideas? Thanks so much.