I want to show that if I make the angles between basis vectors $e_1,...,e_n\in\Bbb R^n$ smaller, then the angles between the dual basis vectors $e_1^*,...,e_n^*\in\Bbb R^n$ become larger.
More preceisly:
Question: Suppose I have two bases $e_1,...,e_n\in\Bbb R^n$ and $\bar e_1,...,\bar e_n\in\Bbb R^n$ so that $\measuredangle(e_i,e_j)\le \measuredangle(\bar e_i,\bar e_j)$ for all $i,j\in\{1,...,n\}$, then does for the dual bases $e_1^*,...,e_n^*\in\Bbb R^n$ and $\bar e_1^*,...,\bar e_n^*\in\Bbb R^n$ hold $\measuredangle(e_i^*,e_j^*)\ge \measuredangle(\bar e_i^*,\bar e_j^*)$ for all $i,j\in\{1,...,n\}$?
More technically, if I have a partial order on the bases of $\Bbb R^n$ given by component wise comparison of angles, I want to know whether the map $\{\text{basis}\}\to\{\text{dual basis}\}$ inverts this partial order.