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: •Σas (Eur)" (Eur)" j=1 where p > 1 and 1 1 + P q 1. Cauchy-Schwarz inequality: En ≤ (Clear) ( Minkowski inequality: + k=1 m=1 Σ12 m=1 Σ + (Ex-er)'s (Eur)² - (Eur)". j=1 where p > 1. k=1 m=1 Problem 5: Compactness in Functional Spaces Problem Statement: Consider the space C([0, 1], R) of continuous real-valued functions on the interval [0, 1] equipped with the supremum norm ||-||- Tasks: a) Arzelà-Ascoli Theorem: State the Arzelà-Ascoli Theorem and use it to characterize the compact subsets of C ([0, 1], R). b) Application: Let F be the set of functions f(x)=sin(na) for n € N. Determine whether F is relatively compact in C([0, 1], R). Justify your answer using the Arzelà-Ascoli Theorem. c) Compact Operator: Define the differentiation operator D : C¹ ([0, 1], R) → C([0,1], R) by Df=f'. Investigate whether D is a compact operator. d) Visualization: For a sequence of functions in C([0, 1], R) that converges uniformly, plot their graphs to illustrate uniform convergence. Conversely, provide a sequence that does not have a uniformly convergent subsequence and visualize it.