In this post, the answer proves that there is a bijection between the following definitions for the tangent space
Suppose $M$ is a smooth manifold (not necessarily a submanifold of $\mathbb{R} ^n$) and $p \in M$. Define \begin{align*} C_p (M) := \left\{ \gamma : (-\epsilon , \epsilon ) \rightarrow M \text{ differentiable with } \gamma (0) = p \right\}. \end{align*} Define an equivalence relation on $C_p (M)$ by \begin{align*} \alpha \sim \beta \iff (\phi \circ \alpha )' (0) = (\phi \circ \beta )' (0) \text{ for some chart } \phi. \end{align*} The tangent space $T_p M$ at $p$ is defined by $T_p M = C_p (M) / \sim $.
and
Let $p \in U \subseteq M$, where $U$ is open. The tangent space of $M$ at $p$ is the vector space $$ T_p M = \{ \, \text{derivations on $U$ at $p$} \, \} \equiv \text{Der}_p(C^\infty(U)). $$ The subscript $p$ tells us the point at which we are taking the tangent space.
where
A derivation on an open subset $U \subseteq M$ at $p \in U$ is a linear map $X: C^\infty(U) \to \mathbb{R}$ satisfying the Leibniz rule $$ X(fg) = f(p) X(g) + g(p) X(f). $$
I have no problems with the proof. However, I am aware that we can attach a vector space structure to each space, with the first being
For each $(h,U)$ with $p \in U$, $T_p M$ is a vector space with operations $$ v + w : = h_* ^{-1} (h_* (v) + h_* (w)) \quad \text{and} \quad \lambda v : = h_* ^{-1} (\lambda h_* (v)), $$ with $h_* : [\gamma ]_{~} \mapsto (h \circ \gamma )' (0)$.
and for the second being the structure being inherited by the dual space in which it is contained.
Question: Are these vector spaces isomorphic? I tried to prove it but to no avail and I am wondering if this is even true.
