Builder notation for vectors

Question

I need to write a vector that contains a scalar for each random variable in a set $S$. I would like something similar to set-builder notation to be able to define vectors as in

$$\mathbf v = (v_s | s \in S).$$

Would that be acceptable or is there something more appropriate to build vectors?

One problem I see straightway is that we have no ordering over the elements of $S$ there.

What I currently do is to define $v_s$ and then say $\mathbf v$ is the vector that "collects" all $v_s$, but that does not seem very rigorous either.

Answer 1

What you could do is give a basis $$ \{ e_{s} : s \in S \} $$ for your vector space and then define $$ (v_{s} : s \in S) = \sum_{s} v_{s} e_{s}. $$ Alternatively, you could just use the sum instead of defining new notation. Lastly, you could consider $v$ as a function from $S$ to your field of scalars (presumably the real numbers). That is, $v: s \mapsto v_{s}$. It depends on what you're trying to use this for.

Edit: The thing about rigor is that it can be a bit tough to pin down what it means exactly. A better goal would just to aim for being completely unambiguous. In this case, the structure of a vector space makes your life easier. Regardless of your choice of notation, if you can show 1) that your vector is an element of some vector space and 2) that you can unambiguously determine its components with respect to a single basis of that space, then you are good.

For instance, the set of functions $f: X \to \mathbb{R}$ forms a vector space under the addition $(f + g)(x) = f(x) + g(x)$ for any finite set $X$. So if you take $X = S$, you get my second example. Alternatively, per your suggestion in the comments, if you take $ X = \{ 1, 2, \dots n \} $ as some index set, then you can represent your vector as a function $f: X \to \mathbb{R}$ so long as you also define an injective function $g: X \to S$ to index the random variables (it must be injective so that no two distinct random variables have the same index).

Answer 2

I think that notation would be acceptable. Just define it before you first use it. The alternative is to index the random variables in some arbitrary way, thus imposing an arbitrary order and then using clumsy essentially meaningless subscripts to refer to them.

Answer 3

To describe a vector, we usually describe its entries. For example, a $3$-dimensional vector might be described as $v_1=2,\,v_2=-3,\,v_3=4$, or another vector would be described by (say) $v_s=f(s),\,s\in S$.