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

[Math Amateur]

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:
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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 ...

 

[castor28]

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.

 

[math771]

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.