Why is Supremum of a.u Less Than or Equal to r||a||_2?

[peterlam]

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!

 

[chiro]

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").

 

[peterlam]

Sorry. I mean Euclidean norm.

Thanks!

 

[Landau]

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 $\ell^2$ norm, except that here we are simply in a finite-dimensional space. In more conventional notation,

$$\sup\{\left<x,y\right>\ :\ \|y\|\leq r\}=r\|x\|.$$

@peterlam: to show LHS $\leq$ RHS use Cauchy-Schwartz .