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I want to exhibit smooth structure and vector bundle structure on $L_{alt}^k(TM)=\bigcup_{p\in M} L_{alt}^k(T_pM)$ where $M$ is a manifold of dimension $n$ and $L_{alt}^k(T_pM)$ is the set of all $k$-th alternating multilinear maps on $T_pM$.

I can guess that for a chart $(U,x)$ of $M$ , I have to construct some map form $\bigcup_{p\in U} L_{alt}^k(T_pM)$ to $U×\Bbb R^{\binom{n}{k}}$.I don't know deep theory of vector bundle, so some rigorous proof or reference that contains all explanation on this topic, will be helpful to me.

Thanks in advance.

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    $\begingroup$ Lemma 10.6 in Lee's Smooth Manifold might help. $\endgroup$ – Sou Oct 14 '18 at 20:19

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