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I am reading Andrew Browder's book: "Mathematical Analysis: An Introduction" ... ...

I am currently reading Chapter 8: Differentiable Maps and am specifically focused on Section 8.1 Linear Algebra ...

I need some help in fully understanding some remarks by Browder after Lemma 8.4 pertaining to the "operator norm" Lemma 8.4 ... The relevant text including Lemma 8.4 reads as follows:

View attachment 9373

View attachment 9374

Near the end of the above text we read the following:

" ... ... We leave it to the reader to show that the sup and inf in (8.2) and (8.3) are actually max and min, i.e. that they are actually attained ... ... "

Could someone please demonstrate rigorously that the sup and inf in (8.2) and (8.3) are actually max and min, i.e. that they are actually attained ... ...

Help will be much appreciated ...

Peter

I am currently reading Chapter 8: Differentiable Maps and am specifically focused on Section 8.1 Linear Algebra ...

I need some help in fully understanding some remarks by Browder after Lemma 8.4 pertaining to the "operator norm" Lemma 8.4 ... The relevant text including Lemma 8.4 reads as follows:

View attachment 9373

View attachment 9374

Near the end of the above text we read the following:

" ... ... We leave it to the reader to show that the sup and inf in (8.2) and (8.3) are actually max and min, i.e. that they are actually attained ... ... "

Could someone please demonstrate rigorously that the sup and inf in (8.2) and (8.3) are actually max and min, i.e. that they are actually attained ... ...

Help will be much appreciated ...

Peter