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. 1 (3)·(3) ( Problem 11: Hahn-Banach Theorem and Its Applications Problem Statement: Let X be a real normed vector space, and let Y be a subspace of X. Suppose f : Y → R is a bounded linear functional. Tasks: a) Hahn-Banach Extension: State the Hahn-Banach Theorem. Use it to extend f to a bounded linear functional f: X→ R such that ƒ agrees with f on Y and ||f|| = ||||- b) Separation of Convex Sets: Using the Hahn-Banach Theorem, prove that two disjoint convex sets in X can be separated by a hyperplane, provided one of them is open. c) Dual Space Identification: Identify the dual space (C([a, b], R))* using the Riesz Representation Theorem. d) Visualization: Illustrate the separation of two disjoint convex sets in R² with a hyperplane (a line) as guaranteed by part (b). Provide a sketch showing the convex sets and the separating line. ΣΚ Holder inequality: ls&P j=1 where p > 1 and + 1 1 1. P q IGP)" Cauchy-Schwarz inequality: ≤12 Minkowski inequality: + k-1 (+1) where p > 1. m=1 m=1