Definition:Product of Affine Spaces

Definition
Let $\mathcal E,\mathcal F$ be affine spaces with difference spaces $E, F$ respectively.

Let $\mathcal G = \mathcal E \times \mathcal F$ be the cartesian product of the sets $\mathcal E,\mathcal F$.

Let $G = E \times F$ be the direct product of the vector spaces $E,F$.

Define sum and difference operations $+ : \mathcal G \times G \to \mathcal G$ and $- : \mathcal G \times \mathcal G \to G$ by, for all $p,p' \in \mathcal E$ and $q,q' \in \mathcal F$:


 * $\left({p, q}\right) + \left({p', q'}\right) := \left({p + p', q + q'}\right)$
 * $\left({p, q}\right) - \left({p', q'}\right) := \left({p - p', q - q'}\right)$

Then the set $\mathcal G$ together with the vector space $G$ and the operations $+,-$ is called the product of the affine spaces $\mathcal E$ and $\mathcal F$.

Also see

 * Product of Affine Spaces is Affine Space