Consider a function $F:A\to W$, with $A\subseteq V$. $F$ is said to be differentiable at $a\in A$ if there exists a $T\in \text{Hom}(V,W)$ such that $$ \Delta F_a(\xi)=T(\xi)+o(\xi). $$ $\Delta F_a(\xi)=F(a+\xi)- F(a)$.
I want to understand what ($a+\xi$) means in this context.
It cannot be that this is a vector space $+$, as $A$ is not a vector space. The natural question is this: Is an open set, $A$, of a vector space, $V$, an affine space modeled on $V$?
In this case $a+\xi$ would be the affine map $\alpha : A\times V \to A$. My feeling, however, is that the answer is no because $\xi$ need be "sufficiently small" so that $a+\xi \in A$, and $\xi$ "sufficiently" small doesn't consitute a vector space.