The exploration of normalized solutions for the nonlinear Schrödinger equation can be traced back to the study of standing wave solutions with prescribed L 2 -norm. Jürgen Moser firstly introduced the concept of normalized solutions in the study of regularity properties of solutions to elliptic partial differential equations (elliptic PDEs). Specifically, he used normalized sequences of functions to prove regularity results for solutions of elliptic equations, which was a significant contribution to the field. Inequalities developed by Emilio Gagliardo and Louis Nirenberg played a crucial role in the study of PDE solutions in L p spaces. These inequalities provided important tools and background for defining and understanding normalized solutions.