I have an issue with the proof of theorem 4-5 in Spivak's Calculus on Manifolds. The theorem states
4-5 Theorem. The set of all
$\varphi_{i_1}\wedge\cdots\wedge\varphi_{i_k}\quad 1\leq i_1<i_2<...<i_k\leq n$
is a basis for $\Lambda^k(V)$.
Here $\varphi_1,...,\varphi_n$ is the dual basis corresponding to some basis of $V$, and $\Lambda^k$ is the space of alternating k-tensors on $V$.
Now, in the proof Spivak invokes theorem 4-1 to write for any $\omega\in\Lambda^k(V)$,
$$\omega = \sum_{i_1,...,i_k}a_{i_1,...,i_k}\varphi_{i_1}\otimes\cdots\otimes\varphi_{i_k}. $$
Here i think the sum is supposed to be over k-tuples $1\leq i_1<...<i_k\leq n$, since the notation $\sum_{i_1,...,i_k=1}^n$ isn't used as it was previously in the chapter. But what justifies writing $\omega$ in this form? Theorem 4-1 only states that the set of all
$$\varphi_{i_1}\otimes\cdots\otimes\varphi_{i_k}\qquad 1\leq i_1,...,i_k\leq n $$
forms a basis for $\mathcal{J}^k(V)$, the space of k-tensors on $V$. How is the sum missing so many terms?
Edit: If we assume what is meant by the author is $\sum_{i_1,...,i_k}=\sum_{i_1,...,i_k=1}^n$, then the proof doesn't work. The argument continues by writing $$\omega = Alt(\omega) = \sum_{i_1,...,i_k} a_{i_1,...,i_k}Alt(\varphi_{i_1}\otimes\cdots\otimes\varphi_{i_k})=\sum_{i_1,...,i_k} a_{i_1,...,i_k} C\varphi_{i_1}\wedge\cdots\wedge \varphi_{i_k}$$ for a constant $C$ (which I think should just equal $1/k!$ ?). But this doesn't show that $\omega$ is the linear combination of the tensors in the statement of the theorem!