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Why do we define $l_{2}$ to be the space of real sequences $(x_{k})$ such that $\sum_{k=1}^\infty\vert x_{k} \vert^{2}$ converges instead of the space of sequences such that $\sum_{k=1}^\infty x_{k}{}^{2}$ converges? $\vert x_{k}\vert^{2}=x_{k}^{2}$ after all...

This question may be silly, but I want to be sure there's nothing mysterious going on here.

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    Sometimes, one considers sequences with $x_k \in \Bbb C$. – 2017-02-16
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    Usually, not sometimes! – 2017-02-16
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    Ok, the definition in my book is only for real numbers. Thank you. – 2017-02-16
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    @Simoes They probably didn't want you to be confused later. For real numbers, you are right that the absolute value is not needed. – 2017-02-16

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