Understanding Andrew Browder's Prop 8.7: Operator Norm and Sequences

In summary, Browder's book provides an introduction to the mathematics of analysis, including linear algebra, differentiable maps, and calculus. However, the last assertion of Proposition 8.7 is left unproven, and the book does not provide a formal and rigorous demonstration of the assertion.
  • #1
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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
 

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  • #2
I'm a little confused. Isn't that what the proof in the book is supposed to do?
 
  • #3
Ackbach said:
I'm a little confused. Isn't that what the proof in the book is supposed to do?
Hi Ackbach ... ...

Hmmm ... can only say I agree with you ...

But Browder leaves the assertion unproven ...

I can only assume that Browder thinks the proof is obvious and trivial ... but I am having problems formulating a proof .. so I hope that someone can help ...

PeterEDIT: I note in passing that in relation to the assertion \(\displaystyle \| I - L^{ -1 } \| \leq \| I - L \| ( 1 - \| I - L \|)^{-1} \)we have that \(\displaystyle \| I - L \| ( 1 - \| I - L \|)^{-1} = \frac{ \| I - L \| }{ ( 1 - \| I - L \|) }\)\(\displaystyle = \| I - L \| + \| I - L \|^2 + \| I - L \|^3 + \ ... \ ... \ ... \)
But I cannot see how to use this in the proof ... but it might be helpful :) ... Peter
 
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  • #4
Browder has shown during the proof of Proposition 8.7 that $\|I-S\| \leqslant \dfrac t{1-t}$. But $S=L^{-1}$ and $t = \|T\| = \|I-L\|$. So that inequality becomes $\|I-L^{-1}\| \leqslant \dfrac {\|I-L\|}{1- \|I-L\|}$.
 
  • #5
Opalg said:
Browder has shown during the proof of Proposition 8.7 that $\|I-S\| \leqslant \dfrac t{1-t}$. But $S=L^{-1}$ and $t = \|T\| = \|I-L\|$. So that inequality becomes $\|I-L^{-1}\| \leqslant \dfrac {\|I-L\|}{1- \|I-L\|}$.
... Hmmm ... I should have seen that .

Appreciate the help, Opalg ...

Peter
 

1. What is Andrew Browder's Prop 8.7?

Andrew Browder's Prop 8.7 is a mathematical proposition that deals with the operator norm and sequences in functional analysis. It was proposed by mathematician Andrew Browder in his book "Introduction to Function Analysis" published in 1969.

2. What is the operator norm?

The operator norm is a mathematical concept used to measure the size or magnitude of an operator in functional analysis. It is defined as the maximum value that the operator can attain on a given space.

3. How is the operator norm related to sequences?

In the context of Andrew Browder's Prop 8.7, the operator norm is used to study sequences of operators. The proposition states that if a sequence of operators converges in the operator norm, then it also converges pointwise on a dense subset of the space.

4. What are some applications of Prop 8.7?

Prop 8.7 has various applications in functional analysis, such as in the study of bounded linear operators, compact operators, and spectral theory. It is also used in the proof of important theorems, such as the Banach-Steinhaus theorem and the uniform boundedness principle.

5. Is Prop 8.7 still relevant in modern mathematics?

Yes, Prop 8.7 is still relevant in modern mathematics and is widely used in various areas of functional analysis. It has also been extended and generalized in different ways, making it a fundamental concept in the field.

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