Different Forms of Operator Norms

[maritimer]

Homework Statement

I was given the first definition but am not sure how to get the last 3

Homework Equations

N/A

The Attempt at a Solution

I tried taking sup (with restriction being llxll=1) on both sides of the inequation,
llF(x)ll=<llFll llxll, but would eventually end up with sup(llF(x)ll ; llxll=1) = sup(llF(x)ll ; llxll=1).
Is this the right way to start deriving those 3 equations or am I supposed to derive from the very first equation?

Thanks for the help.

 

[mighty2000]

Thank you for your question. To derive the last three equations, you can use the definition of supremum, which states that for a set S, the supremum of S is the least upper bound of S. In this case, the set S is the set of all values of llF(x)ll for llxll=1. So, the supremum of S, sup(llF(x)ll ; llxll=1), is the smallest value of llF(x)ll that is greater than or equal to all values of llF(x)ll for llxll=1.

Using this definition, you can derive the last three equations as follows:

1. sup(llF(x)ll ; llxll=1) = sup(llF(x)ll) : This equation follows directly from the definition of supremum. The supremum of a set is equal to the supremum of the set itself.

2. sup(llF(x)ll ; llxll=1) >= llF(x)ll : This equation also follows from the definition of supremum. Since the supremum is the least upper bound, it must be greater than or equal to any value in the set.

3. sup(llF(x)ll ; llxll=1) = llF(0)ll : This equation follows from the fact that for llxll=1, x=0. So the set S contains only one value, llF(0)ll. Therefore, the supremum of S is equal to this value.

I hope this helps. Let me know if you have any further questions.