Let \(V\) be a finite dimensional inner product space . Then for every covariant tensors \(\alpha\in T^kV^*\) and a unit vector \(v\in V\) we have
\begin{equation*} \lvert \alpha(v,\ldots ,v)\rvert^2\le \langle \alpha, \alpha\rangle. \end{equation*}
Proof
Choose an orthonormal basis \(e_1, e_2, \ldots \) with \(e_1=v\). Then
\begin{equation*} \alpha(v,\ldots ,v)=\alpha_{1,\ldots ,1} \end{equation*}and therefore
\begin{equation*} \lvert \alpha(v,\ldots ,v)\rvert^2=\alpha_{1,\ldots ,1}^2\le \sum_{i_1,\ldots ,i_k} \alpha_{i_1,\ldots ,i_k}^2=\langle \alpha, \alpha\rangle. \end{equation*}