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 .

 

[CellCoree]

The supremum of a dot u, subject to the 2-norm of u being less than or equal to r, represents the maximum possible value that the dot product of a and u can have while still satisfying the condition that the 2-norm of u is less than or equal to r. This means that the supremum is the largest possible value of the dot product, given the constraint on the 2-norm of u.

Now, the 2-norm of a vector is defined as the square root of the sum of the squared entries of the vector. In other words, it represents the length of the vector. Similarly, the 2-norm of u represents the length of the vector u.

In order for the supremum to be r times the 2-norm of a, it means that the dot product of a and u must be maximized when the length of u is equal to r times the length of a. This can be visualized geometrically as the vector u being stretched or scaled to have a length of r times the length of a.

Since the dot product of two vectors is equal to the product of their lengths multiplied by the cosine of the angle between them, when the length of u is r times the length of a, the dot product will be maximized. This is because the cosine of any angle is always less than or equal to 1.

Therefore, the supremum of the dot product of a and u, subject to the 2-norm of u being less than or equal to r, is equal to r times the 2-norm of a, as the dot product is maximized when the length of u is r times the length of a.