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: ΣΚ ·E (Eur)' (E)". j=1 where p > 1 and Cauchy-Schwarz inequality: G j=1 Minkowski inequality: + where p > 1. 1 + 1 1 1. m=1 Σ k=1 m=1 + m=1 P Problem 21: Uniform Boundedness Principle Problem Statement: The Uniform Boundedness Principle is a cornerstone of functional analysis. Tasks: a) Uniform Boundedness Principle Statement: State the Uniform Boundedness Principle (Banach- Steinhaus Theorem). b) Application to Pointwise Convergence: Use the Uniform Boundedness Principle to show that if a sequence of bounded linear operators {T} on a Banach space X converges pointwise to a bounded operator T', then sup, ||T|| < 0. c) Counterexample Without Boundedness: Provide an example where pointwise convergence fails to imply uniform boundedness if the Uniform Boundedness Principle's conditions are not met. d) Visualization: Illustrate the Uniform Boundedness Principle by showing a family of linear operators on R² where each operator is bounded, and their pointwise limits also maintain boundedness. Include diagrams of operator actions.