Explore the intricacies of the Hahn-Banach theorem in this 33-minute lecture by Hajime Ishihara, presented as part of the Hausdorff Trimester Program on Types, Sets and Constructions. Delve into the theorem's various forms, including the continuous extension theorem, separation theorem, and dominated extension theorem. Examine the constructive approaches to these theorems, focusing on Bishop's approximate separation theorem and its corollary. Investigate how geometric properties of Banach spaces, such as uniform convexity and Gateaux differentiability of the norm, contribute to exact versions of the separation and continuous extension theorems for normable linear functionals on nonseparable spaces. Learn about subderivatives and Gateaux derivatives of convex functions, and explore the relationship between subdifferentiability and the separation and dominated extension theorems. Discover a constructive version of the Mazur theorem using the constructive Baire theorem, along with its corollaries. The lecture covers a comprehensive syllabus, including introductory concepts, examples, bounded linear mappings, normable linear functionals, located sets, and various constructive approaches to key theorems in functional analysis. Read more