Help Understanding Andrew Browder's Proof of Proposition 8.14 from Math Analysis

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In summary, the conversation is discussing a specific proposition, 8.14, in Andrew Browder's book "Mathematical Analysis: An Introduction". The focus is on understanding the proof of the proposition and specifically, the statement made by Browder regarding limits. The summary also includes a correction to a mistake made by Browder in his argument.
  • #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 8.2 Differentials ... ...

I need yet further help in fully understanding the proof of Proposition 8.14 ...

Proposition 8.14 reads as follows:
View attachment 9409

In the above proof by Browder, we read the following:" ... ... For any \(\displaystyle v \in \mathbb{R}^n\), and \(\displaystyle t \gt 0\) sufficiently small, we find (taking \(\displaystyle h = tv\) above) that \(\displaystyle L(tv) + r(tv) \leq 0\), or \(\displaystyle Lv \leq r(tv)/t\), so letting \(\displaystyle t \to 0\) we have \(\displaystyle Lv \leq 0\). ... ... Now ... the above quote implies that

\(\displaystyle \lim_{ t \to 0 } \frac{ r(tv) }{ t } = 0\) ... ... But why exactly (formally and rigorously) is this the case ... ... ?I note that we have that \(\displaystyle \lim_{ h \to 0 } \frac{ r(h) }{ |h| } = \lim_{ t \to 0 } \frac{ r(tv) }{ tv } = 0\)... but this is (apparently anyway) not exactly the same thing ... we need to be able to demonstrate rigorously that\(\displaystyle \lim_{ h \to 0 } \frac{ r(h) }{ |h| } = \lim_{ t \to 0 } \frac{ r(tv) }{ t } = 0\) ... ... ... but how do we proceed to do this ...?Hope someone can help ... ...

Peter

==============================================================================EDIT:

Just noticed that in the above quote, Browder argues that \(\displaystyle L(tv) + r(tv) \leq 0\) implies that \(\displaystyle Lv \leq r(tv)/t\) ... ... ... BUT ... i suspect he should have written \(\displaystyle L(tv) + r(tv) \leq 0\) implies that \(\displaystyle Lv \leq - r(tv)/t\) ... ..... however ... in either case ... when we let \(\displaystyle t \to 0\) we get the same result ... namely \(\displaystyle Lv \leq 0 \)... ==============================================================================
 

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  • #2
Peter said:
In the above proof by Browder, we read the following:" ... ... For any \(\displaystyle v \in \mathbb{R}^n\), and \(\displaystyle t \gt 0\) sufficiently small, we find (taking \(\displaystyle h = tv\) above) that \(\displaystyle L(tv) + r(tv) \leq 0\), or \(\displaystyle Lv \leq r(tv)/t\), so letting \(\displaystyle t \to 0\) we have \(\displaystyle Lv \leq 0\). ... ... Now ... the above quote implies that

\(\displaystyle \lim_{ t \to 0 } \frac{ r(tv) }{ t } = 0\) ... ... But why exactly (formally and rigorously) is this the case ... ... ?I note that we have that \(\displaystyle \lim_{ h \to 0 } \frac{ r(h) }{ |h| } = \lim_{ t \to 0 } \frac{ r(tv) }{ tv } = 0\)... but this is (apparently anyway) not exactly the same thing ... we need to be able to demonstrate rigorously that\(\displaystyle \lim_{ h \to 0 } \frac{ r(h) }{ |h| } = \lim_{ t \to 0 } \frac{ r(tv) }{ t } = 0\) ... ... ... but how do we proceed to do this ...?
This is another case where you have to distinguish between fixed and variable vectors. Here, $v$ is fixed, but $tv$ varies, and goes to $0$ as $t\to0$. So \(\displaystyle \lim_{t\to0}\frac{r(tv)}{t|v|} = 0\), and then you can multiply by the fixed nonzero scalar $|v|$ to get \(\displaystyle \lim_{t\to0}\frac{r(tv)}{t} = 0\).

Peter said:
Just noticed that in the above quote, Browder argues that \(\displaystyle L(tv) + r(tv) \leq 0\) implies that \(\displaystyle Lv \leq r(tv)/t\) ... ... ... BUT ... i suspect he should have written \(\displaystyle L(tv) + r(tv) \leq 0\) implies that \(\displaystyle Lv \leq - r(tv)/t\) ... ..... however ... in either case ... when we let \(\displaystyle t \to 0\) we get the same result ... namely \(\displaystyle Lv \leq 0 \)...
Absolutely correct! Browder has omitted a minus sign. But his conclusion is correct.
 
  • #3
Opalg said:
This is another case where you have to distinguish between fixed and variable vectors. Here, $v$ is fixed, but $tv$ varies, and goes to $0$ as $t\to0$. So \(\displaystyle \lim_{t\to0}\frac{r(tv)}{t|v|} = 0\), and then you can multiply by the fixed nonzero scalar $|v|$ to get \(\displaystyle \lim_{t\to0}\frac{r(tv)}{t} = 0\).Absolutely correct! Browder has omitted a minus sign. But his conclusion is correct.

HI Opalg ...

Your post was most helpful to me ...

Peter
 

1. What is Proposition 8.14 in Math Analysis?

Proposition 8.14 in Math Analysis is a mathematical statement that states the existence of a unique solution to a differential equation under certain conditions.

2. Who is Andrew Browder and why is his proof important?

Andrew Browder was a renowned mathematician who made significant contributions to the field of functional analysis. His proof of Proposition 8.14 is important because it provides a rigorous and logical explanation for the existence of a unique solution to a differential equation, which is a fundamental concept in mathematics.

3. What is the difficulty level of understanding Andrew Browder's proof?

The difficulty level of understanding Andrew Browder's proof may vary depending on the individual's mathematical background and familiarity with the concepts of functional analysis. However, it is generally considered to be a challenging proof that requires a solid understanding of advanced mathematical concepts.

4. Are there any prerequisites for understanding Andrew Browder's proof?

Yes, there are some prerequisites for understanding Andrew Browder's proof. These include a strong foundation in calculus, linear algebra, and real analysis. Familiarity with concepts such as Banach spaces, compactness, and continuity is also recommended.

5. How can I improve my understanding of Andrew Browder's proof?

To improve your understanding of Andrew Browder's proof, it is recommended to first review the prerequisites and make sure you have a solid understanding of the underlying concepts. It may also be helpful to read through the proof multiple times, consult with a mentor or colleague, and work through related practice problems to solidify your understanding.

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