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- Jun 22, 2012

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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:

My 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:

My 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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