MHB 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:
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 ...

 

[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.