The objective of this paper is to rigorously define the Kadec-Klee property for modular spaces endowed with a sequential convergence structure, and to demonstrate that this property leads to the normal structure in such spaces. Consequently, we establish that the Kadec-Klee property defined herein implies the corresponding fixed point property for these spaces. These results are new in the modular space setting. Furthermore, given that the examined class of spaces encompasses Banach spaces, modular function spaces, and various other types, our theory offers a comprehensive, unified framework for exploring the interconnections between the Kadec-Klee property, normal structure, and the fixed point property.
Wojciech M. Kozƚowski (Wed,) studied this question.