While going through some aspects of differential geometry from the book by Amari and Nagaoka on "Methods of information geometry", I came across the following definition for a metric connection:
Regarding the notation, $\mathcal{T}(S)$ denotes the set of all $C^{\infty}$ vector fields on the manifold $S$. While the right hand side of $(1.65)$ is a sum of two $C^{\infty}$ functions on $S$, the left hand side seems to be a vector field $Z$ multiplied by a $C^{\infty}$ function, resulting in vector field as a whole. This is clearly not possible.
Can anyone kindly help me parse $(1.65)$ correctly? In particular, I would like to know how to interpret the quantity on the left hand side of $(1.65)$.