Definition:Product of Affine Spaces

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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 $\mathcal F$.

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