Franklin's Notes

Tangent space

Given a point $x$ in an affine space $X$ with difference function $d$, we can define a vector space structure for the space ${x}\times X$ from the structure of $T$, by defining addition and scalar multiplication via the equations ...using the fact that the partially-applied map $d_x$ is always a bijection. The fact that these definitions satisfy the vector-space axioms follow from the definitional properties of the difference map $d$ and the fact that $T$ is a vector space.

The space defined by the above is called the tangent space to $X$ at $x$ and is denoted $T_x X$. Its vectors are called tangent or bound vectors at $x$, whereas the vectors in $T$ are called free. Additionally, $d_x$ is called a freeing map, and its inverse is called a binding map.