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 $C^\infty\left({V, \R} \right)$ to be the set of all smooth mappings $f : V \to \R$.

Then a tangent vector $X_m$ at $m$ is a linear mapping $X_m : C^\infty\left({V, \R} \right) \to \R$ satisfying the Leibniz law:
 * $\displaystyle X_m \left( {f g} \right) = X_m \left( {f}\right) \, g\left({m} \right) +f \left({m} \right) \, X_m \left( {g} \right)$