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: (E) (E)'. j=1 1 where p > 1 and + P 1 Σ Cauchy-Schwarz inequality: ² ykis (r) (u)! j=1 m Minkowski inequality: (Σ5+1") (Exter)'s (Eur)²+(Eur) Σ m=1 Problem 18: Compactness in Infinite-Dimensional Spaces Problem Statement: Compactness behaves differently in infinite-dimensional Banach spaces compared to finite- dimensional ones. Tasks: a) Riesz's Theorem: State Riesz's Theorem regarding compact subsets of infinite-dimensional Hilbert spaces. b) Sequential Compactness: Explain why the unit ball in an infinite-dimensional Hilbert space is not compact in the norm topology. c) Compact Operators: Define compact operators and provide an example of a compact operator on 12. d) Visualization: Illustrate why the unit ball in (2 is not compact by showing a sequence with no convergent subsequence. Include a diagram of points on the unit sphere spreading out without accumulation. where p > 1.