Norm question

  • Thread starter peterlam
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  • #1
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Consider a and u are vector of n entries,
why the supremum of a dot u subject to the 2-norm of u is less than or equal to r equals r times 2-norm of a, i.e. sup{a.u | ||u||_2 <=r} = r ||a||_2?

How can I work out that?

Thank you!
 

Answers and Replies

  • #2
chiro
Science Advisor
4,790
132
Consider a and u are vector of n entries,
why the supremum of a dot u subject to the 2-norm of u is less than or equal to r equals r times 2-norm of a, i.e. sup{a.u | ||u||_2 <=r} = r ||a||_2?

How can I work out that?

Thank you!

Can you please give a definition of the two-norm as I haven't encountered it before (1 and infinity norm, and L^p norms but not "2-norm").
 
  • #3
16
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Sorry. I mean Euclidean norm.

Thanks!
 
  • #4
Landau
Science Advisor
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Can you please give a definition of the two-norm as I haven't encountered it before (1 and infinity norm, and L^p norms but not "2-norm").
This would be the [itex]\ell^2[/itex] norm, except that here we are simply in a finite-dimensional space. In more conventional notation,

[tex]\sup\{\left<x,y\right>\ :\ \|y\|\leq r\}=r\|x\|.[/tex]

@peterlam: to show LHS [itex]\leq[/itex] RHS use Cauchy-Schwartz .
 

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