Definition:Product of Affine Spaces

Definition

Let $\EE, \FF$ be affine spaces with difference spaces $E, F$ respectively.

Let $\GG = \EE \times \FF$ be the Cartesian product of the sets $\EE, \FF$.

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

Define sum and difference operations $+ : \GG \times G \to \GG$ and $- : \GG \times \GG \to G$ by, for all $p, p' \in \EE$ and $q, q' \in \FF$:

$\tuple {p, q} + \tuple {p', q'} := \tuple {p + p', q + q'}$

$\tuple {p, q} - \tuple {p', q'} := \tuple {p - p', q - q'}$

Then the set $\GG$ together with the vector space $G$ and the operations $+, -$ is called the product of the affine spaces $\EE$ and $\FF$.

Also see

• Product of Affine Spaces is Affine Space