This course offers a comprehensive and rigorously structured exposition of Topology and
Functional Analysis, designed for students enrolled in the Master 1 program in Mathematics
with a specialization in Mathematical Analysis and Applications at Djilali Bounaama University
– Khemis Miliana. It covers the foundational pillars of general topology—continuity,
compactness, connectedness, and separation axioms—before advancing to the core of functional
analysis: Banach and Hilbert spaces, linear operators, duality, and the landmark theorems that
shape the discipline, including Hahn–Banach, Open Mapping, Closed Graph, and Uniform
Boundedness.
Special attention is given to the weak topology and the weak* topology, two indispensable
tools that reveal the subtle interplay between convergence, compactness, and duality in
infinite-dimensional spaces. These topologies are not merely theoretical curiosities; they are
essential in the study of partial differential equations, variational methods, optimization, and
the mathematical formulation of quantum mechanics. Mastery of these concepts empowers students
to navigate advanced research literature and to engage confidently with modern analytical
techniques across pure and applied mathematics.
Through clear exposition, carefully selected examples, and fully worked exercises, this text
aims to bridge the gap between abstract theory and practical understanding—equipping learners
not only to succeed in their coursework but also to build a durable foundation for future
academic or professional endeavors in mathematics and related fields.
- معلم: Mihoub BOUDERBALA