### 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**.

# geometry

# vector-space

# affine-space

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