User:Geometry dude/Definition:Tangent Space

Definition
Let $M$ be a smooth manifold.

Let $m \in M$ be a point.

Let $V$ be an open neighborhood of $m$.

Define $\map {C^\infty} {V, \R}$ to be the set of all smooth mappings $f : V \to \R$.

Then a tangent vector $X_m$ at $m$ is a linear mapping $X_m : \map {C^\infty} {V, \R} \to \R$ satisfying the Leibniz Law:
 * $\map {X_m} {f g} = \map {X_m} f \map g m + \map f m \map {X_m} g$