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

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In the discussion about Lemma 1.1.7 from "Multidimensional Real Analysis I" by Duistermaat and Kolk, a clarification is sought regarding the notation $$x^{(k)}$$. It is explained that the parentheses indicate an index rather than a power, distinguishing it from the coordinates, which are represented by subscripts. The notation refers to a sequence of vectors, specifically $$x^{(1)}, \ldots, x^{(\ell)}$$. This distinction is crucial for understanding the lemma's context and application. Overall, the conversation emphasizes the importance of notation in mathematical texts for accurate interpretation.
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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 $$x^{ (k) }$$ ... but what does $$x^{ (k) }$$ mean ... ? ... is it $$x$$ to the power $$k$$ ... ? ... ... 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 ...
 
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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.
 

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