# Multidimensional Real Analysis - Duistermaat and Kolk, Lemma 1.1.7 ....

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In summary, Multidimensional Real Analysis is a branch of mathematics that focuses on studying functions, sequences, and series of multiple variables. It is covered in the book "Multidimensional Real Analysis" written by Dutch mathematicians Duistermaat and Kolk. Lemma 1.1.7 in the book provides a specific result related to multidimensional functions and is useful in proving other theorems and results. However, the book is not suitable for beginners and is more suited for students with a strong foundation in real analysis and multivariable calculus.
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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 Lemma 1,1,7 (ii) ...

Duistermaat and Kolk"s Lemma 1.1.7 reads as follows:
View attachment 7640
View attachment 7641
In the above Lemma part (ii), Duistermaat and Kolk refer to $$\displaystyle x^{ (k) }$$ ... but what does $$\displaystyle x^{ (k) }$$ mean ... ? ... is it $$\displaystyle x$$ to the power $$\displaystyle k$$ ... ? ... ... what exactly does it mean ... indeed, why are there parentheses around k ... ... ... and further, how do we interpret or make sense of part (ii) ...Help will be appreciated ...

Peter*** NOTE ***

I have searched, but cannot find an explanation by Duistermaat and Kolk of a notation in which there are parentheses around the exponent ...

Last edited:
Hi Peter,

I think that this is just an index: you have $\ell$ vectors $x^{(1)},\ldots,x^{(\ell)}$. The index is written as a superscript, because subscripts are used for the coordinates.

Hi Peter,

I am not familiar with the specific notation used by Duistermaat and Kolk, but in general, the notation x^{(k)} can mean a few different things. It could mean x to the power of k, as you mentioned, but it could also mean the kth derivative of x, or the kth element of a sequence or vector x.

In order to understand part (ii) of Lemma 1.1.7, it would be helpful to know the context in which x^{(k)} is being used. Can you provide some more information or context about the lemma or the book you are reading? That may help clarify the notation and make sense of part (ii).

In general, if you are unsure about a specific notation or concept in a book, it may be helpful to look for other resources or explanations online. You could also try reaching out to the authors directly for clarification.

I hope this helps. Let me know if you have any other questions.

## 1. What is Multidimensional Real Analysis?

Multidimensional Real Analysis is a branch of mathematics that deals with the study of functions, sequences, and series of several variables.

## 2. Who are Duistermaat and Kolk?

Duistermaat and Kolk are two Dutch mathematicians who wrote the book "Multidimensional Real Analysis", which is a comprehensive text on the subject.

## 3. What is Lemma 1.1.7 in "Multidimensional Real Analysis - Duistermaat and Kolk"?

Lemma 1.1.7 is a mathematical statement in the book "Multidimensional Real Analysis" that provides a specific result related to the study of multidimensional functions.

## 4. How is Lemma 1.1.7 useful in Multidimensional Real Analysis?

Lemma 1.1.7 is useful in Multidimensional Real Analysis as it helps in proving other theorems and results related to the subject.

## 5. Is "Multidimensional Real Analysis - Duistermaat and Kolk" suitable for beginners?

No, "Multidimensional Real Analysis - Duistermaat and Kolk" is a more advanced text and is suitable for students with a strong foundation in real analysis and multivariable calculus.

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