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. 1 (3)·(3) ( Problem 11: Hahn-Banach Theorem and Its Applications Problem Statement: Let X be a real normed vector space, and let Y be a subspace of X. Suppose f : Y → R is a bounded linear functional. Tasks: a) Hahn-Banach Extension: State the Hahn-Banach Theorem. Use it to extend f to a bounded linear functional f: X→ R such that ƒ agrees with f on Y and ||f|| = ||||- b) Separation of Convex Sets: Using the Hahn-Banach Theorem, prove that two disjoint convex sets in X can be separated by a hyperplane, provided one of them is open. c) Dual Space Identification: Identify the dual space (C([a, b], R))* using the Riesz Representation Theorem. d) Visualization: Illustrate the separation of two disjoint convex sets in R² with a hyperplane (a line) as guaranteed by part (b). Provide a sketch showing the convex sets and the separating line. ΣΚ Holder inequality: ls&P j=1 where p > 1 and + 1 1 1. P q IGP)" Cauchy-Schwarz inequality: ≤12 Minkowski inequality: + k-1 (+1) where p > 1. m=1 m=1

Traffic Control  The accompanying figure shows the flow of downtown traffic during the rush hours on a typical weekday. Each avenue can handle up to 1500 vehicles/hour without causing congestion, whereas the maximum capacity of each street is 1000 vehicles/hour. The flow of traffic is controlled by traffic lights at each of the six intersections. (A) Suggest two possible traffic-flow patterns that ensure no traffic congestion. (B) Suppose the traffic flow along 9th Street between 5th and 6th Avenues, x6, is restricted due to sewer construction. What is the minimum permissible traffic flow along this road that will not result in traffic congestion?

Please help, thank you.

***PLEASE HELP WITH THE FOLLOWING AND PLEASE PROVIDE HANDWRITTEN WORK ON PAPER*** (9) Determine if               S = (7 - 4x + 4x2 , 6 + 2x - 3x2,20 - 6x + 5x2) is linearly independent or dependent

Linear Algebra. What is a vector space axiom?

Suppose y1 ( x), y2 ( x), y3 ( x) are three different functions of x. The vector space they span could have dimension 1, 2, or 3. Give an example of y1, y2, y3 to show each possibility.

i understand part 1,3 and 4 can someone explain part 2 to me

A certain country uses a progressive tax system. The amount of tax consists of a linear part proportional to the income and a nonlinear part depending on the income by a power law. The total amount of tax is determined by the formula T(W)=aW+(bW+c)p, where W is the income; p is the exponent, a,b,c are some positive numbers. At what level of income the tax rate will be minimal?

Please solve problem D 2. Thanks.

Please do Parts D,E,F please.    Thanks