Theorem: Tangent Space is Vector Space

Let $M$ be a smooth manifold of dimension $n \in \N$. 
Let $m \in M$ be a point. 
Let $\struct {U, \kappa}$ 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: 
:$\set {\valueat {\dfrac \partial {\partial \kappa^i} } m : i \in \set {1, \dotsc, n} }$
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 $\map {C^\infty} {V, \R}$ be the set of smooth mappings $f: V \to \R$. 
Let $X_m, Y_m \in T_m M$.
Let $\lambda \in \R$.
Then, by definition of tangent vector and Equivalence of Definitions of Tangent Vector:
:$X_m, Y_m$ are linear transformations on $\map {C^\infty} {V, \R}$.  
Hence $\paren {X_m + \lambda Y_m}$ are also linear transformations. 
Therefore, it is enough to show that $X_m + \lambda Y_m$ satisfies the Leibniz law. 
Let $f, g \in \map {C^\infty} {V, \R}$. 
Then:
{{begin-eqn}}
{{eqn | l = \map {\paren {X_m + \lambda Y_m} } {f g}
      | r = \map {X_m} {f g} + \lambda \map {Y_m} {f g}
      | c = {{Defof|Linear Transformation}}
}}
{{eqn | r = \map {X_m} f \map g m + \map f m \map {X_m} g + \lambda \paren {\map {Y_m} f \map g m + \map f m \map {Y_m} g}
      | c = Leibniz law for $X_m, Y_m$ 
}}
{{eqn | r = \map {\paren {X_m + \lambda Y_m} } f \map g m + \map f m \map {\paren {X_m + \lambda Y_m} } g
      | c = reordering summands
}}
{{end-eqn}}
It follows that:
:$X_m + \lambda Y_m \in T_m M$
Hence $T_m M$ is a real vector space. 
Again, by definition of tangent vector and Equivalence of Definitions of Tangent Vector:
:for all $X_m \in T_m M$ there exists a smooth curve:
::$\gamma: I \subseteq \R \to M$
:where $\map \gamma 0 = m$ such that:
{{begin-eqn}}
{{eqn | l = \map {X_m} f
      | r = \valueat {\map {\frac {\map \d {f \circ \gamma} } {\d \tau} } \tau} {\tau \mathop = 0}
}}
{{eqn | r = \valueat {\map {\frac {\map \d {f \circ \kappa^{-1} \circ \kappa \circ \gamma} } {\d \tau} } \tau} {\tau \mathop = 0}
      | c = $f \circ \kappa^{-1} \circ \kappa = f$, as $\kappa$ is a homeomorphism, in particular a bijection.
}}
{{eqn | r = \sum_{i \mathop = 1}^n \valueat {\map {\frac {\map \partial {f \circ \kappa^{-1} } } {\partial \kappa^i} } {\map {\kappa \circ \gamma} \tau} \map {\frac {\map \d {\kappa^i \circ \gamma} } {\d \tau} } \tau} {\tau \mathop = 0}
      | c = Chain Rule for Real-Valued Functions
}}
{{eqn | r = \sum_{i \mathop = 1}^n \map {\frac {\map \d {\kappa^i \circ \gamma} } {\d \tau} } 0 \map {\frac {\map \partial {f \circ \kappa^{-1} } } {\partial \kappa^i} } {\map {\kappa \circ \gamma} 0}
      | c = rearranging
}}
{{eqn | r = \sum_{i \mathop = 1}^n \map {\frac {\map \d {\kappa^i \circ \gamma} } {\d \tau} } 0 \map {\frac {\map \partial {f \circ \kappa^{-1} } } {\partial \kappa^i} } {\map \kappa m}
      | c = as $m = \map \gamma 0$
}}
{{end-eqn}}
We define:
:$X^i_m := \map {\dfrac {\map \d {\kappa^i \circ \gamma} } {\d \tau} } 0$
and as above: 
:$\valueat {\map {\dfrac \partial {\partial \kappa^i} } m} f := \map {\dfrac {\map \partial {f \circ \kappa^{-1} } } {\partial \kappa^i} } {\map \kappa m}$
Therefore: 
:$\ds \map {X_m} f = \map {\paren {\sum_{i \mathop = 1}^n X^i_m \valueat {\dfrac \partial {\partial \kappa^i} } m} } f$
{{iff}}:
:$\ds X_m = \sum_{i \mathop = 1}^n X^i_m \valueat {\frac \partial {\partial \kappa^i} } m$
Hence:
:$\set {\valueat {\dfrac \partial {\partial \kappa^i} } m: i \in \set {1, \dotsc, n} }$
forms a basis. 
Hence, by definition of dimension of vector space:
:$\dim T_m M = n = \dim M$
This completes the proof. 
{{qed}}
Category:Smooth Manifolds
Category:Tangent Spaces
Category:Vector Spaces
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