TITLE: Differential forms dimensions
QUESTION [0 upvotes]: So I am studying differential geometry and I am struggling to understand how 1.18 comes from 1.17? If $p=1$ it must be $n$.

REPLY [0 votes]: We have
$$
\Lambda V = \bigoplus_{p=0}^\infty \Lambda^p V \implies \dim \Lambda V = \sum_{p=0}^\infty \dim \Lambda^p V
$$
but you know that
$$
\dim \Lambda^p V = 0\quad \text{for} \quad p>n
$$
It follows from $(1.17)$ that
$$
\dim \Lambda V = \sum_{p=0}^n \binom{n}{p} = 2^n \tag{1.18}
$$
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