The Concept of Measures of Noncompactness in Banach Spaces

This article is a survey, and its aim is to provide a concise introduction to the topic of measures of noncompactness and measures of weak noncompactness in Banach spaces. These measures constitute a useful tool in nonlinear analysis, for example, in studies on the existence of solutions to nonlinear differential and integral equations. Recently, they have also been applied to the analysis of infinite systems of such equations. Throughout the paper, particular attention is given to highlighting the symmetry that exists between these concepts. Some open problems are also included at the end of the paper.