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: P ·S (Sp) (B) j=1 1 Σ where p > 1 and 1 1 + Р q 1. Σ m=1 Cauchy-Schwarz inequality: P •₤()() Minkowski inequality: j=1 k=1 m=1 (Ex₁+)'s (E)² + (₤²) + where p > 1. Σ m=1 Problem 17: Weak and Weak Convergence* Problem Statement: In Banach spaces, notions of convergence beyond norm convergence are essential. Tasks: a) Definitions: Define weak convergence and weak* convergence in Banach and dual spaces, respectively. b) Banach-Alaoglu Theorem: State the Banach-Alaoglu Theorem and explain its significance in the context of weak* convergence. c) Eberlein-Smulian Theorem: State and prove the Eberlein-Smulian Theorem, which relates weak compactness and sequential weak compactness in Banach spaces. d) Visualization: Provide an example in R2 where a sequence converges weakly but not strongly. Illustrate the sequence and its weak limit with a graph.