Product space

A Cartesian product of vector spaces with componentwise operations

Product space

Let $X_1,\dots,X_m$ be vector spaces over the same field $K$ . Their Cartesian product

X:=X_1\times\cdots\times X_m

becomes a vector space over $K$ by defining, for $x=(x_1,\dots,x_m)$ and $y=(y_1,\dots,y_m)$ in $X$ and $\alpha\in K$ ,

x+y:=(x_1+y_1,\dots,x_m+y_m),\qquad \alpha x:=(\alpha x_1,\dots,\alpha x_m).

This vector space is called the product space (or direct product) of $X_1,\dots,X_m$ .

Examples:

$\mathbb{R}^n$ is the product of $n$ copies of $\mathbb{R}$ .
If $X=Y\times Z$ , then the subsets $Y\times\{0\}$ and $\{0\}\times Z$ are subspaces whose sum is all of $X$ .
For function spaces, $C[a,b]\times C[a,b]$ is a product space of pairs of continuous functions.