I was wondering if someone could clarify this for me. In my lecture notes my lecturer keeps writing "Let $\{\textbf{v}_1,\ldots, \textbf{v}_n\}\in \mathbb{R}^m$ be a collection of vectors $\dots$" and was wondering if this isn't an abuse of notation since each $\textbf{v}_i\in \mathbb{R}^m$ and thus $\{\textbf{v}_1,\ldots, \textbf{v}_n\}\subset \mathbb{R}^m$ as this set itself is not an element of $\mathbb{R}^m$.
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$\begingroup$ Yes, it's an abuse of notation (for the reason you gave). $\endgroup$ – Daniel Fischer♦ Mar 22 '14 at 15:39
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$\begingroup$ Too much abuse, to be honest. $\endgroup$ – Git Gud Mar 22 '14 at 15:40
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Yes, that is abuse of (at least sloppy) notation, for exactly the reasons you gave. Especially so with the set clammers. It would be a lot more acceptable with the more normal:
$$v_1, \dots, v_n \in \Bbb R^n$$
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1$\begingroup$ This even has less characters, no reason not to use this over the other one. $\endgroup$ – Git Gud Mar 22 '14 at 15:43
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1$\begingroup$ Another reason for not writing the set brackets is that one often doesn't want to exclude the possibility that some of the $v_i$s are equal. $\endgroup$ – Henning Makholm Mar 22 '14 at 16:11
