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

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In summary, 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, and this can be shown by using Cauchy-Schwartz and the definition of the 2-norm as the Euclidean norm.
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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!
 
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  • #2
peterlam said:
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
Sorry. I mean Euclidean norm.

Thanks!
 
  • #4
chiro said:
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 .
 
  • #5


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.
 

1. Why is the supremum of a.u less than or equal to r||a||_2?

The supremum of a set is the least upper bound of the set, and it represents the maximum value that the elements of the set can attain. In this case, a.u refers to the dot product of two vectors, and ||a||_2 is the norm of vector a. The supremum of a.u is less than or equal to r||a||_2 because the dot product of two vectors can never be greater than the product of their norms. This is a fundamental property of vector spaces and can be mathematically proven.

2. What is the significance of the supremum in this equation?

The supremum plays a crucial role in understanding the upper bound of the dot product between two vectors. It helps us determine the maximum possible value that the dot product can attain and is essential in various mathematical proofs and applications.

3. How does this equation relate to vector norms?

The equation relates to vector norms as ||a||_2 represents the Euclidean norm, which is a type of vector norm. The supremum of a.u is less than or equal to r||a||_2 because the dot product is bounded by the product of the vector norms, which is a property of all vector norms.

4. Can you provide an example to illustrate this equation?

Suppose we have two vectors, a = [2, 3] and u = [4, 5]. The dot product of these vectors is a.u = (2*4) + (3*5) = 8 + 15 = 23. The Euclidean norm of vector a is ||a||_2 = sqrt(2^2 + 3^2) = sqrt(4 + 9) = sqrt(13). Therefore, according to the equation, the supremum of a.u is less than or equal to r||a||_2, which is 23 ≤ r*sqrt(13). This example demonstrates the relationship between the supremum and vector norms.

5. How is this equation used in real-life applications?

The equation is commonly used in various fields such as engineering, physics, and computer science. It is used to determine the maximum possible value of a physical quantity, to prove mathematical theorems, and to optimize algorithms. For example, the supremum of the dot product can be used to find the maximum possible magnitude of a force in physics or the maximum possible distance between two points in a computer program.

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