We find the best constants in inequalities relating the standard norm, the dual norm, and the norm ‖𝑥‖(𝑝,𝑠)∑∶=inf{𝑘‖𝑥(𝑘)‖𝑝,𝑠}, where the infimum is taken over all finite representations ∑𝑥=𝑘𝑥(𝑘) in the classical Lorentz sequence spaces. A crucial point in this analysis is the concept of level sequence, which we introduce and discuss. As an application, we derive the best constant in the triangle inequality for such spaces.