I want to define a map, which maps a vector $x \in \mathbb{R}^n$ to a new vector $y \in \mathbb{R}^m$, such that every entry of $y$ is a fixed entry of $x$ (duplicates are allowed). My current approach looks like this:
Let $I \subseteq \{1, \dots, n\}^m$. We define $$x \mapsto (x_i)_{i \in I}$$
but I'm not sure, whether this captures the ordering of the elements correctly. I think that I could fix this issue by writing
$$x \mapsto (x_{i_j})_{j = 1, \dots, m}$$
but this seems too messy. Now, I am asking whether there es a simple way to define this map? Does the first definition work or am I missing something? Something like $x_I$ is great, but I don't think that this is standard notation and sadly it is not contained in the multi-index notation as well (even though I think it would be a nice extension).
My understanding is that you want to specify some indices (say $1, 3, 4, 1$ for example) and then get the map (if $n = 5$) $$ (x_1, x_2, x_3, x_4, x_5) \mapsto (x_1, x_3, x_4, x_1). $$ Your second version captures this.
The first version does not work: If $I \subseteq \{1, \dots, n\}^m$, then an element $i \in I$ is a sequence $(i_1, \dots, i_m)$ and $x_i$ does not make sense.
Depending on how much you use this operation, you might consider defining $x_I := (x_{i_1}, \dots, x_{i_m})$ for $I = (i_1, \dots, i_m) \in \{ 1, \dots, n \}^m$ (note: element, not subset). This notation is not standard (as you seem to know) but introducing custom notation is fine as long as you clearly state what it means; it certainly seems like a natural choice.