Tangent Space is Vector Space

Theorem
Let $M$ be a smooth manifold of dimension $n \in \N$.

Let $m \in M$ be a point.

Let $\left({U, \kappa}\right)$ be a chart with $m \in U$.

Let $T_m M$ be the tangent space at $m$.

Then $T_m M$ is a real vector space of dimension $n$, spanned by the basis:


 * $\displaystyle \left\{{\frac \partial {\partial \kappa^i }_{\restriction_m} : i \in \left\{{1, \dots, n}\right\}  }\right\}$

that is the set of partial derivatives with respect to the $i$th coordinate function $\kappa^i$ evaluated at $m$.

Proof
Let $V$ be an open neighborhood of $m$ with $V \subseteq U \subseteq M$.

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

Let $X_m, Y_m \in T_m M$ and $\lambda \in \R$.

Then, by Definition:Tangent Vector and Equivalence of Definitions of Tangent Vector,

$X_m, Y_m$ are linear mappings on $C^\infty \left( { V, \R } \right)$.

Hence $\left( {X_m + \lambda Y_m } \right)$ are also linear mappings.

Therefore, it is enough to show that $X_m + \lambda Y_m$ satisfies the Leibniz law.

Let $f, g \in C^\infty \left( { V, \R } \right)$.