Let \(V\) be a \(n\)-dimensional vector space. If \((\varepsilon^i)\) is any dual basis , the for each positive integer \(k\le n\), the collection of \(k\)-covectors
\begin{equation*} \mathcal{E} = \{\varepsilon^I: I \text{ is an increasing multi-index} \} \end{equation*}is a basis of alternating \(k\)-tensors \(\Lambda^k(V^*)\).
Remarks
- \(\dim \Lambda^k(V^*) = \binom{n}{k}\).
- We abbreviate the summation over all increasing multi-indices by the primed sum sign, i.e.
\begin{equation*}
\sum_{I}' \alpha_I\varepsilon^I := \sum_{1\le i_1<\ldots