I am reading this introduction to Mechanics and the definition it gives (just after Proposition 1.1.2) for an affine subspace puzzles me.
I cite:
A subset $B$ of a $\mathbb{R}$-affine space $A$ modelled on $V$ is an affine subspace if there is a subspace $U$ of $V$ with the property that $y−x \in U$ for every $x,y \in B$
It later says that this definition is equivalent to to the usual one, namely that of closeness under sum with elements of a $U$, but it seems to me that there is a problem with the first definition. Just imagine the usual $\mathbb{R}^2$ plane as an affine space modeled on $\mathbb{R}^2$. According to this definition the subset $\{(0,0);(0,1)\}$ is an affine subspace, while this is not so according to the usual definition of an affine subspace. Is there an error in the book?