Week 3 Math 1530 Advanced Calculus 1 - RUBIN - Fall ’03SCHEDULE: Homework from Week 3 is due Wednesday, September 17.• Wednesday, September 10th: start Section 1.7• Friday, September 12th: finish Section 1.7, start Section 2.1• Monday, September 15th: finish Section 2.1, Section 2.2TOPICS:Section 1.7: Norms, Inner Products, and Metrics - In this section, we define normedspaces, inner product spaces, and metric spac es in general. The book gives various examples ofthese spaces, equipped with appropriate norms, inner products, and metrics, respectively. Tocheck that these examples really satisfy the definitions, one simply needs to show that they satisfyeach of the properties for the appropriate type of space.Reading Objectives: After reading Section 1.7, students should be able to:• List the properties of normed spaces, inner product spaces, and metric space s.• Understand the relationship between norms, inner products, and metrics; that is, which ofthese can be defined from which other of these?• Give an example of the usefulness of the Cauchy-Schwarz inequality.homework:1. pg. 98. # 102. pg. 98, # 12a3. pg. 100, # 30Section 2.1: Open Sets - The essential idea in this section is that an open set A is one inwhich for eve ry point x in A, there is a disk containing x that is entirely c ontained within A.Reading Objectives: After reading Section 2.1, students should be able to:• Give examples of sets that are open or not open (in IR and in IRn).• Determine whether particular examples of sets, such as {(x, y) ∈ IR2: x + y < 3} or{(x, y) ∈ IR2: xy > 1}, are open or not.• Characterize which unions and intersec tions of open sets are open.homework:1. Write out the details for Exercise 2.1.5 on pg. 108. That is, for A ⊂ IR open and B ⊂ IR,define AB = {xy ∈ IR : x ∈ A and y ∈ B} and then do the following.a) Determine the form of those sets B for which AB is not open.b) Prove that for all other B, AB is open.Hint: There are special closed sets B for which AB is not open. There are other closedsets B for which the result depends on A. For all other B, AB is actually open.12. pg. 146, #20a.Section 2.2: Interior of a Set - The interior of a set is the largest open subset o f the set,given by the c ollection of all points in A which lie inside open sets entirely contained in A.Reading Objectives: After reading Section 2.2, students should be able to:• Find the interior of a se t, such as {(x, y) ∈ IR2: x + y ≤ 3}.homework:1. Give an example of A, B ⊂ IRnfor som e n for which int(A ∪ B) 6= int(A) ∪ int(B). Be sureto justify your answer.2. pg. 145, #