Linear transformations between normed linear spaces

In summary, when working with linear transformations between normed linear spaces, the norm of the transformation is defined as the supremum of the norm of the transformation for all possible values of x where the norm of x is less than or equal to 1. However, when trying to show that this is also equal to the supremum of the norm for all possible values of x where the norm of x is exactly equal to 1, it is often assumed to be obvious. To understand this, one can manipulate the function to fit within a closed unit sphere and work with x divided by its norm.
  • #1
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Hi, ok I'm working with linear transformations between normed linear spaces (nls)

if T :X -> Y nls's is a linear transformation, we define the norm of T, ||T||: sup{||T|| : ||x||<=1}

I want to show that for X not = {0}
||T||: sup{||T|| : ||x|| = 1} frustratingly the books all assume that this step is obvious... I don't see how.

Intuitively I can see that is true, using the fact that (I think) T is a bounded function, and we can manipulate things to make the whole function be contained within a closed unit sphere...
Have always had a mental block with inf's and sups...don't know why...
 
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  • #2
Work with x/||x||. It will be obvious.
 
  • #3


Thank you for sharing your thoughts on linear transformations between normed linear spaces. To show that ||T||: sup{||T|| : ||x|| = 1} is true, we can use the definition of supremum. By definition, the supremum of a set S is the least upper bound of S. In this case, the set S is {||T|| : ||x|| = 1}. This means that for any value of ||T||, there exists at least one x such that ||x|| = 1 and ||T|| is less than or equal to the supremum. This shows that ||T||: sup{||T|| : ||x|| = 1} is indeed a valid definition of the norm of T.

To further understand this, we can also consider the fact that ||T|| is the maximum value that T can take on for any x in X. This means that for any x with ||x|| = 1, ||T(x)|| must be less than or equal to ||T||. This is because ||T(x)|| is a scalar multiple of ||T||, and the norm of a scalar multiple is the absolute value of the scalar multiplied by the norm of the original vector. Therefore, ||T|| is indeed the supremum of {||T|| : ||x|| = 1}.

I understand that it can be frustrating when books assume certain steps to be obvious, but it's important to remember that different people have different ways of understanding and explaining concepts. It's always helpful to approach things from different perspectives and ask questions when something is not clear. I hope this explanation helps clarify the definition of the norm of a linear transformation between normed linear spaces for you.
 

1. What is a linear transformation between normed linear spaces?

A linear transformation between normed linear spaces is a function that preserves vector addition and scalar multiplication, and also preserves the norm of vectors. This means that the length and direction of vectors are maintained after the transformation.

2. How do you determine if a linear transformation between normed linear spaces is continuous?

A linear transformation between normed linear spaces is continuous if and only if it is bounded. This means that there exists a constant C such that the norm of the transformed vector is less than or equal to C times the norm of the original vector.

3. Can a linear transformation between normed linear spaces be invertible?

Yes, a linear transformation between normed linear spaces can be invertible if it is both one-to-one and onto. This means that there exists a unique inverse transformation that can map the transformed vectors back to their original form.

4. What is the difference between a linear transformation and an isomorphism between normed linear spaces?

A linear transformation between normed linear spaces is a function that preserves vector addition, scalar multiplication, and norm, while an isomorphism is a bijective linear transformation. This means that an isomorphism is not only a linear transformation, but it also has a unique inverse transformation.

5. How are the properties of a linear transformation between normed linear spaces affected by the choice of norms?

The properties of a linear transformation between normed linear spaces are not affected by the choice of norms. This is because all norms are equivalent in finite-dimensional spaces, meaning that they lead to the same topology and therefore, the same properties for linear transformations.

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