two-sided invariant linear functionals

例句：

1. 1. Two-sided invariant linear functionals arecrucial for the stability of quantum systems.
   双向不变线性功能对于量子系统的稳定至关重要。
2. 2.In this study, we investigate the two-sided invariant linear functionalsto understand their impact on the system's behavior.
   在这项研究中，我们研究了双向不变线性功能，以了解它们对系统行为的影响。
3. 3. The researchers foundthat these two-sided invariantlinear functionals could be used to predictthe future behavior of the system.
   研究人员发现，这些双向不变线性功能可以用来预测系统的未来行为。