I am reading Information geometry by "Amari" and I am having a problem understanding how the author defined the following linear map,
Intuitively, defining an affine connection on a manifold $S$ means that for each point $p\in S$ and its "neighbor" $p'$, we define a linear one-to-one mapping between $T_p$ and $T_{p'}$. Here we call $p'$ a neighbor of $p$ if, given a coordinate system $[\xi^i]$ of $S$, the difference between the coordinates of $p$ and $p'$, $d\xi^i=\xi^i(p')-\xi^i(p)$. when construed as a first-order infinitesimal, is sufficiently small that we may ignore the second-order infinitesimals $(d\xi^i)(d\xi^j)$. Below we shall introduce the notion of affine connections in an intuitive manner using infinitesimals.
In order to establish a linear mapping $\prod_{p,p'}$ between $T_P$, and $T_{p'}$, we must specify, for each $j\in\{1,2,......,n\}$, how to express $\prod_{p,p'}((\partial_{j})_p)$ in terms of a linear combination of $\{(\partial_1)_{p'},....,(\partial_n)_{p'}\}$ where $\partial_j=\frac{\partial}{\partial\xi^j}$. Let us def assume that the difference between $\prod_{p,p'}((\partial_{j})_p)$ and $(\partial_j)_{p'}$ , is infinitesimal. and that it may be expressed as a linear combination of $\{d\xi^1,....,d\xi^n\}$. Then we have
$\prod_{p,p'}((\partial_{j})_p)=(\partial_j)_p'-d\xi^i(\Gamma_{ij}^{k})_{p}(\partial_k)_{p'}$
where $\{(\Gamma_{ij}^{k})_{p};i,j,k=1,...,n\}$ are $n³$ real numbers which depend on the point $p$ and $p'$ in $S$. I am stuck here and do not get any idea about how this map is defined. If someone explains it that will be a great help.