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. yes(Eur) (E). Holder inequality: < Cauchy-Schwarz inequality: j=1 where p > 1 and + 1 1 1. P q Σ ΣΕΡ Minkowski inequality: (Σ16 + 1/º)*: where p > 1. k=1 m + Σ m=1 Problem 29: Reflexivity and Weak Topologies Problem Statement: Understanding reflexivity involves exploring weak topologies on Banach spaces. Tasks: a) Weak Topology Definition: Define the weak topology on a Banach space X. b) Characterizing Reflexivity: Prove that a Banach space X is reflexive if and only if every bounded sequence in X has a weakly convergent subsequence. c) Examples and Non-Examples: Provide examples of reflexive and non-reflexive spaces, explaining how the weak topology behaves in each case. d) Visualization: Illustrate weak convergence in 2 by showing a sequence converging weakly but not strongly. Include diagrams highlighting convergence behavior.