Using the notation of families to denote vectors

Question

Is it common to use the notation for families in the context of vector spaces? For example, is it common to write $\{x_i\}\in R^n$? Alternatively, is there any other similar notation that avoids writing $(x_1,x_2,\dots,x_n)\in R^n$?

Answer 1

To me, the expression $\{x_i\}\in\mathbb R^n$ is meaningless. $\{x_i\}$ is a singleton, i.e., a set with one element, which is $x_i$. This set cannot be an element of $\mathbb R^n$, since $\mathbb R^n$ contains tuples, i.e. elements of the form $(x_1, \dots x_n)$.

If you want a simple way of denoting an arbitrary element of $\mathbb R^n$, you can simply use $x$, and it is usually implicitly accepted that $x=(x_1,\dots, x_n)$.