Notation for set of concatenated vectors?

Question

A standard way of writing concatenation of two vectors $x\in \mathbb{R}^{m}$ and $y\in \mathbb{R}^n$ is $[x^\top, y^\top]^\top\in \mathbb{R}^{m+n}$. Let $x\in X\subset \mathbb{R}^n$ and $y\in Y \subset \mathbb{R}^m$. Then, $[x^\top, y^\top]^\top\in Z$, where $Z:= \{[x^\top, y^\top]^\top : x\in X, y\in Y\}$. Is there any standard notation for the set $Z$?

Answer 1

The vector space $X\oplus Y=\{\,x +y\mid x\in X, y\in Y\,\}$ of formal sums$^1$ is isomorphic to the vector space of concatenations via $$f\colon X\oplus Y \to Z, \quad x+y \mapsto [x^\top,y^\top]^\top$$

So in terms of vector spaces, there is not really a difference between $Z$ and $X\oplus Y$. I think this is the reason why there is no well established notation for this. But you can just use $x+y$. If you want to emphasize that the elements come from different vector spaces (to differentiate from the inner addition of the vector space), you can use $x\oplus y$.

You could borrow the notation from other areas, but I think most readers won't understand notations like "$X\cdot Y$" or "$xy$".

$^1$ if you just add two vector spaces, this is the same as $X\times Y=\{\,(x,y)\mid x\in X, y\in Y\,\}$.

Answer 2

According to wiki the string concatenation can be written as follows

$$Z = \{ xy : x \in X, y \in Y\}.$$

Also, take a look to these questions: Is there a common symbol for concatenating two (finite) sequences? and String/vector concatenation symbol.