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: Cauchy-Schwarz inequality: Minkowski inequality: j=1 where p > 1 and j=1 + 1 + 1 P q ( 1. ΣΠΙ m=1 (Eur)' (E)' k=1 (+1) where p > 1. < k=1 Σ {" +Στα m=1 Problem 25: Measure Theory and Integration in Functional Analysis Problem Statement: Integration theory plays a vital role in functional analysis, especially in defining LP spaces. Tasks: a) Lebesgue Dominated Convergence Theorem: State the Lebesgue Dominated Convergence Theorem and explain its importance in LP spaces. b) Fubini's Theorem: State Fubini's Theorem and provide a proof for its application in product measure spaces. c) Duality via Integration: Show how integration defines the duality pairing between LP and L. d) Visualization: For simple functions in I2([0, 1]), depict the integral of their product, illustrating the duality pairing. Include graphs of the functions and shaded regions representing the integrals.