I'm reading Fractals Everywhere by Michael Barnsley. On pp. 6-8 [1] he defines a linear space which, he says, "is also called a vector space." However, his definition of a linear space only requires closure under vector addition and scalar multiplication. There is no mention, for example, of additive inverses. (I realize other properties are required.)
He also defines the Riemann Sphere (see section 1.5), and later, in an exercise, asks the reader to show that the Riemann Sphere is not a vector space.
I can see that the Riemann Sphere is not a vector space in the traditional sense because $\infty$ has no additive inverse. But it seems that using Barnsley's definition, it would be a vector space, if we adopt the convention that $\infty + x = \infty$ for all $x \in \mathbb{C}$ and that $a \cdot \infty = \infty$ for all $a \in \mathbb{R}$.
Am I missing something? Thanks.