THE ROLE OF FUNCTIONAL ANALYSIS AND BANACH SPACE TECHNIQUES IN MODERN MATHEMATICAL ANALYSIS AND THEIR APPLICATIONS TO PARTIAL DIFFERENTIAL EQUATIONS

Abstract

This paper explores the significant role of functional analysis and Banach space theory within modern mathematical analysis, particularly in the context of solving partial differential equations (PDEs). Functional analysis, as a bridge between abstract algebraic structures and analytical techniques, provides essential tools for the theoretical and practical resolution of complex problems in various fields of mathematics and physics. Banach spaces, being complete normed vector spaces, serve as fundamental frameworks for analyzing linear and nonlinear operators, ensuring convergence, stability, and existence of solutions. The study also highlights the interplay between theory and application, demonstrating how functional analytic methods help to generalize classical calculus results and make rigorous formulations possible for infinite-dimensional problems, especially those arising in PDEs. The research underlines the pedagogical and methodological implications of integrating functional analytic thinking into the mathematical curriculum at the tertiary level in Uzbekistan.