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: j=1 where p 1 and + Cauchy-Schwarz inequality: j=1 Minkowski inequality: + where p > 1. 1 1 P q (Em)" m=1 Ση 12 k=1 m=1 (ΣK+DP)'s (ERP)² + Σm Problem 33: Functional Calculus for Operators Problem Statement: Functional calculus allows the application of functions to operators in a Banach or Hilbert space. Tasks: a) Definition: Define the functional calculus for bounded linear operators on a Hilbert space, particularly using the spectral theorem. b) Polynomial Functional Calculus: Show how to apply polynomial functions to a bounded self- adjoint operator and verify the properties. c) Continuous Functional Calculus: Extend the polynomial functional calculus to continuous functions using the Stone-Weierstrass Theorem. d) Visualization: For a diagonal operator on ², illustrate how applying a function f transforms the operator by applying to each diagonal entry. Include diagrams showing the original and transformed operators.