# Notation for set of concatenated vectors?

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 $$Z$$?

• Why not $X\times Y$? – Babelfish Dec 2 '19 at 9:13
• $z\in X\times Y$ can be interpreted as a tuple $(x,y)$, which is another type of object, not a vector? – Angelos Dec 2 '19 at 9:15

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 $$X\times Y=\{\,(x,y)\mid x\in X, y\in Y\,\}$$.

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.

• I think a notation like $xy$ for two vectors $x\in \mathbb R^m$ and $y\in \mathbb R^n$ would be highly confusing. – Babelfish Dec 2 '19 at 12:44
• @Babelfish, I agree, this is for strings. For that reason I referred to very similar discussions on math and tex SEs. – tortue Dec 2 '19 at 12:45