I was wondering whether the definition of a tangent space is chart dependent or not? Cause to me it seems that they depend on the charts:
Let $p \in M$ where $M$ is a k-dim smooth manifold and $\phi: U(p) \rightarrow \mathbb{R}^k$ a chart where $U(p)$ is an open neighbourhood of $p$ in $M$. Then according to our definition of the tangent space, this one is a space spanned by the directional derivatives $D_v|_p:=\sum_{i=1}^k v_i D_i|_p.$ Now $D_i|_p(f):=D_i(f\circ \phi^{-1})(\phi(p))$ is just defined as the partial derivative in $\mathbb{R}^k$ ($f \in C^{\infty}(M,\mathbb{R})$) via the chart and this is why I guess that the definition of this tangent space, i.e. its basis $\{D_i|_p;i \in \{1,..,k\}\}$ depends on the chart.
If this is true, then also the dual basis (the differential forms ) $dx^i|_p(D_j|_p):= \delta^i_j$ are depedent on the chart and not globally defined on the manifold. Is this correct?