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. •Lo(Bar) (+)) (C&P) Holder inequality: Σ j=1 1 1 where p > 1 and + P q 1. Σ Am=1 ERAIS (ir)" (£) Cauchy-Schwarz inequality: < Minkowski inequality: +|" (Ex)'s (Eur)' (E) where p > 1. + Problem 20: Schauder Basis and Orthonormal Basis Problem Statement: A Schauder basis generalizes the concept of a basis to infinite-dimensional Banach spaces. Tasks: a) Schauder Basis Definition: Define what it means for a sequence {m} in a Banach space X to be a Schauder basis. b) Orthonormal Basis in Hilbert Spaces: Explain how an orthonormal basis in a Hilbert space is a special case of a Schauder basis. c) Existence of Schauder Bases: Prove that spaces have Schauder bases for 1