Weak Derivatives: Definition and Properties

This lecture introduces the concept of weak derivatives, defining them as a set of functions for which specific conditions hold. The instructor explains the properties and characteristics of weak derivatives, emphasizing their importance in various mathematical contexts. Through examples and theoretical explanations, the lecture explores the role of weak derivatives in functional analysis and their relationship with bounded functions. The Ascoli-Arzela theorem is presented as a key result related to weak derivatives, highlighting its significance in understanding the behavior of functions in different spaces. The lecture concludes with practical applications and counterexamples to deepen the understanding of weak derivatives.