Compactness and Uniform Continuity in R^n .... .... D&K Theorem 1.8.15

In summary, the conversation is about Peter seeking help with a specific aspect of the proof of Theorem 1.8.15 in the book "Multidimensional Real Analysis I: Differentiation" by J. J. Duistermaat and J. A. C. Kolk. The conversation includes a request for an explanation and a rigorous proof of the statement that the continuity of the Euclidean norm implies that the limit of the Euclidean distance between two functions is 0. The summary also includes a response from Euge, who explains that this statement is true because of the definition of continuity of the Euclidean norm. Peter expresses his gratitude for the help.
  • #1
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I am reading "Multidimensional Real Analysis I: Differentiation" by J. J. Duistermaat and J. A. C. Kolk ...

I am focused on Chapter 1: Continuity ... ...

I need help with an aspect of the proof of Theorem 1.8.15 ... ...

Duistermaat and Kolk's Theorem 1.8.15 and its proof read as follows:View attachment 7754In the above proof we read the following:

" ... ... The continuity of the Euclidean norm the gives \(\displaystyle \lim_{{k}\to{\infty}} \mid \mid f(x_k) - f(y_k) \mid \mid = 0 \) ... ... "Can someone please explain ... and also show rigorously ... how/why the continuity of the Euclidean norm the gives \(\displaystyle \lim_{{k}\to{\infty}} \mid \mid f(x_k) - f(y_k) \mid \mid = 0 \) ... ... Help will be much appreciated ...

Peter
 
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  • #2
Hi Peter,

It’s almost immediate if you know what continuity of the Euclidean norm means: For any $\mathbf{c}\in \Bbb R^n$ and any sequence $(\mathbf{c}_k)\in \Bbb R^n$ converging to $\mathbf{c}$, $\|\mathbf{c}_k\| \to \|\mathbf{c}\|$. Take $\mathbf{c}_k = f(\mathbf{x}_k) - f(\mathbf{y}_k)$, so $\mathbf{c} = 0$. Since $\|\mathbf{0}\| = 0$, $\|f(x_k) - f(y_k)\| \to 0$.
 
  • #3
Euge said:
Hi Peter,

It’s almost immediate if you know what continuity of the Euclidean norm means: For any $\mathbf{c}\in \Bbb R^n$ and any sequence $(\mathbf{c}_k)\in \Bbb R^n$ converging to $\mathbf{c}$, $\|\mathbf{c}_k\| \to \|\mathbf{c}\|$. Take $\mathbf{c}_k = f(\mathbf{x}_k) - f(\mathbf{y}_k)$, so $\mathbf{c} = 0$. Since $\|\mathbf{0}\| = 0$, $\|f(x_k) - f(y_k)\| \to 0$.
Thanks Euge ...

Appreciate the help ...

Peter
 

FAQ: Compactness and Uniform Continuity in R^n .... .... D&K Theorem 1.8.15

1. What is compactness in R^n and how is it related to uniform continuity?

Compactness in R^n refers to the property of a set being closed and bounded. This means that the set contains all of its limit points and can be enclosed within a finite region. Uniform continuity, on the other hand, is a property of a function where small changes in the input result in small changes in the output. The D&K Theorem 1.8.15 states that a continuous function on a compact set is uniformly continuous.

2. Can you provide an example of a set that is compact in R^n?

One example of a compact set in R^n is a closed interval, such as [0,1]. This set is bounded because it is contained within the region of 0 and 1, and it is also closed because it includes its endpoints. Therefore, it satisfies the criteria for compactness in R^n.

3. How does the D&K Theorem 1.8.15 relate to real-life applications?

The D&K Theorem 1.8.15 is an important concept in mathematical analysis and is used in many real-life applications. For example, it is used in physics to analyze the behavior of continuous systems, in economics to study the stability of markets, and in engineering to design control systems. Understanding compactness and uniform continuity helps in analyzing and predicting the behavior of these systems.

4. How does the D&K Theorem 1.8.15 differ from other theorems related to compactness and uniform continuity?

The D&K Theorem 1.8.15 is specifically focused on the relationship between compactness and uniform continuity in R^n. Other theorems, such as the Heine-Cantor Theorem, also relate compactness and continuity but may have different conditions or apply to different spaces. Additionally, the D&K Theorem 1.8.15 is a direct result of the definition of compactness and uniform continuity, while other theorems may require additional assumptions or proofs.

5. Can a function be uniformly continuous without being defined on a compact set?

Yes, a function can be uniformly continuous without being defined on a compact set. For example, the function f(x)=x^2 is uniformly continuous on the real line, but the real line is not a compact set. The D&K Theorem 1.8.15 only applies to continuous functions on compact sets, but there are other theorems that relate uniform continuity to other types of sets or spaces.

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