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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 yet further help in fully understanding the proof of Proposition 8.7 ...Proposition 8.7 and its proof reads as follows:
View attachment 9399
View attachment 9400My question is as follows:Can someone please demonstrate, formally and rigorously, the last assertion of the above proposition ... ... That is, can someone please demonstrate, formally and rigorously, that ... ... \(\displaystyle \| I - L^{ -1 } \| \leq \| I - L \| ( 1 - \| I - L \|)^{-1} \)
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 yet further help in fully understanding the proof of Proposition 8.7 ...Proposition 8.7 and its proof reads as follows:
View attachment 9399
View attachment 9400My question is as follows:Can someone please demonstrate, formally and rigorously, the last assertion of the above proposition ... ... That is, can someone please demonstrate, formally and rigorously, that ... ... \(\displaystyle \| I - L^{ -1 } \| \leq \| I - L \| ( 1 - \| I - L \|)^{-1} \)
Help will be much appreciated ... ...
Peter
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