Vector Space with Standard Affine Structure is Affine Space

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Theorem

Let $E$ be a vector space.

Let $\struct {\EE, E, +, -}$ be the standard affine structure on $E$.

Then with this structure, $\EE$ is an affine space.


Proof

We are required to show that:

\((1)\)   $:$     \(\ds \forall p, q \in \EE:\) \(\ds p + \paren {q - p} = q \)      
\((2)\)   $:$     \(\ds \forall p \in \EE: \forall u, v \in E:\) \(\ds \paren {p + u} + v = p + \paren {u + v} \)      
\((3)\)   $:$     \(\ds \forall p, q \in \EE: \forall u \in E:\) \(\ds \paren {p - q} + u = \paren {p + u} - q \)      

By definition of the standard affine structure, the addition and subtraction operations are simply those in the vector space $E$.

That is, we want to show that:

\((1)\)   $:$     \(\ds \forall u, v \in E:\) \(\ds u + \paren {v - u} = v \)      
\((2)\)   $:$     \(\ds \forall u, v, w \in E:\) \(\ds \paren {u + v} + w = u + \paren {v + w} \)      
\((3)\)   $:$     \(\ds \forall u, v, w \in E:\) \(\ds \paren {v - u} + w = \paren {v + w} - u \)      

By definition the addition operation on a vector space is commutative and associative.

But all three axioms are immediate consequences of commutativity and associativity.

This concludes the proof.

$\blacksquare$