Definition:Tangent Space

Definition
Let $M$ be a smooth manifold.

Let $m \in M$ be a point.

Then the tangent space at $m$, denoted by $T_m M$, is the set of all tangent vectors at $m$.

Also see

 * Equivalence of Definitions of Tangent Vector
 * Tangent Space is Vector Space
 * Definition:Tangent Bundle