Notation: Address elements of a vector I am dealing with sets of vectors $\big\{x_1, x_2, x_3, \dotsc, x_m \big\}$ from some abstract vector space $\mathcal{V}$. Occasionally, $\mathcal{V}=\mathbb{R}^n$ an I need to address the elements these vectors e.g. sum over all elements of the vector $x_j$. However, the subindex is already used to denote a specific vector.
Is there a common notation address the elements of a vector? E.g. $x_j[i]$ or $x_j^i$ or $x_j^{(i)}$?
 A: In Banach spaces, where we often consider elements of sequence spaces (e.g. $x\in c_0$ so that $x=(x_1, x_2, \ldots)$) the problem of indexing comes up a lot.  Typically when we need to index into a sequence of sequences we use $x_j(n)$ to indicate the $n^{\mathrm{th}}$ component of the $j^{\mathrm{th}}$ element of the master sequence.  I would disagree with Wuestenfux here and say that this is reasonably standard in Banach Space Theory.
Other suggestions, such as raised indices ($x^i_j$) run into problems when you need to consider powers of the series, and multiple sub-indices ($x_{i,j}$) can get confusing and hard to read (there's a Banach space called Schreier space where some of the proofs require considering indices $j$ such that $p_{n_k +1} \leq j \leq p_{n_{k+1}+1}$ which is not only hard to read, but hard to think about!).  Provided you are clear that $x_j$ refers to a sequence and not a function you shouldn't have much difficulty with people understanding $x_j(n)$.
That said, whatever your choice, state it clearly upfront :)
A: Well, there is no common notation here. The policy is to keep the notation as simple as possible.
In your case, I would write $x_{ij}$ for the $j$th component of vector $x_i$.
