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Given a manifold $X\subset R^N$, Guillemin and Pollack define its tangent bundle $T(X)$ as the subset of $X \times \mathbb{R}^N$ given by $$T(X)= \{(x,v) \in X \times \mathbb{R}^N : v \in T_x(X)\},$$ where $T_x(X)$ is the tangent spaces at $x$ on $X$.

But this definition seems contradictory to me: if $\dim X=k$, then $\dim T_x(X)=k$, and a vector $v$ cannot lie simultaneously in $\mathbb R^N$ and in the $k-$dimensional space $T_x(X)$ (maybe unless $k=N$). Please clarify this moment to me.

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    $\begingroup$ In this context the tangent space $T_x(X)$ is a subspace of the tangent space $T_x(\mathbb{R}^N)\cong \mathbb{R}^N$. $\endgroup$ – asdq Apr 9 '18 at 20:35

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