L1-, L2-, Linfty-Norm Proofs -

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In summary, the conversation discusses the proof that for all real numbers x, the L1-norm of x is less than or equal to n times the L-infinity norm of x, and the L2-norm of x is less than or equal to the square root of n times the L-infinity norm of x. The equations for the L1-norm, L2-norm, and L-infinity norm are provided in the conversation. There is also a request for help and a reminder to not expect others to do the homework.
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DeadxBunny
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L1-, L2-, Linfty-Norm Proofs - Please Help!

Homework Statement



Show that ||x||1 < or = n||x||infinity and ||x||2 < or = sqrt(n)*||x||infinity for x exists in the set of all real numbers.


Homework Equations



||x||2 is defined here: http://mathworld.wolfram.com/L2-Norm.html
||x||1 is defined here: http://mathworld.wolfram.com/L1-Norm.html
||x||infinity is defined here: http://mathworld.wolfram.com/L1-Norm.html

Sorry about posting links, but I have no idea how to get all the symbols (like the summation symbol) to show up on the forums!

Thanks in advance for your help!
 
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  • #2
Okay, what have you done? You can't just expect people to do your homework for you! I would also point out that "x exists in the set of all real numbers" makes little sense here. Do you mean that x is in Rn?
 

Related to L1-, L2-, Linfty-Norm Proofs -

What are L1, L2, and Linfty norms?

L1, L2, and Linfty are different types of norms that can be used to measure the magnitude of a vector or a matrix. L1 norm, also known as the Manhattan or taxicab norm, is the sum of the absolute values of the elements of a vector. L2 norm, also known as the Euclidean norm, is the square root of the sum of the squared values of the elements of a vector. Linfty norm, also known as the maximum norm, is the maximum absolute value of the elements of a vector.

Why are L1, L2, and Linfty norms important in proofs?

L1, L2, and Linfty norms are important in proofs because they provide a way to measure the distance or difference between two vectors or matrices. In many cases, these norms can help simplify complex mathematical expressions and make them easier to manipulate in proofs.

How are L1, L2, and Linfty norms used in proofs?

L1, L2, and Linfty norms are used in proofs to show that certain properties hold for a given vector or matrix. For example, in linear algebra, L2 norm is often used to prove convergence of a sequence or to show that a matrix is invertible. In optimization, L1 norm is commonly used to enforce sparsity in solutions.

What are the differences between L1, L2, and Linfty norms?

The main difference between L1, L2, and Linfty norms is the way they measure the magnitude of a vector or matrix. L1 norm takes into account the absolute values of the elements, while L2 norm takes into account the squared values. Linfty norm, on the other hand, only looks at the maximum absolute value. This leads to different properties and applications for each norm.

How do you prove properties of L1, L2, and Linfty norms?

To prove properties of L1, L2, and Linfty norms, one often uses mathematical techniques such as inequalities, triangle inequality, and Cauchy-Schwarz inequality. In some cases, the properties can also be derived from the definitions of the norms themselves. It is important to carefully manipulate the expressions using the properties of the norms to arrive at the desired result.

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