Instructions: *Do not Use AI. (Solve by yourself, hand written preferred) * Give appropriate graphs and required codes. * Make use of inequalities if you think that required. * You are supposed to use kreszig for reference. Holder inequality:Σ&P ·Σavis (Eur) (EN)'. j=1 where p > 1 and 1 + 1 P q 1. m=1 Cauchy-Schwarz inequality: [ { ≤ (²) (~)' Minkowski inequality: ¡inequality: (+1) where p > 1. ΣΙΣΑΙ + ΣΙ m=1 Problem 38: James' Theorem on Reflexivity Problem Statement: James' Theorem provides a characterization of reflexive Banach spaces. Tasks: a) James' Theorem Statement: State James' Theorem regarding the reflexivity of Banach spaces. b) Proof of James' Theorem: Prove one direction of James' Theorem, showing that if a Banach space is reflexive, then every continuous linear functional attains its supremum on the closed unit ball. c) Implications for Optimization: Discuss how James' Theorem influences optimization problems in reflexive Banach spaces. d) Visualization: Illustrate a reflexive Banach space where every continuous linear functional attains its maximum on the unit ball. Provide a graphical example in R².