Studying Rudin's Principles of Mathematical Analysis Through ...

More documents

Recommendations

Info

$Math Department Presentation$

10 CHAPTER 1. THE REAL AND COMPLEX NUMBER SYSTEMS 1.7 Euclidean Spaces 1.7.1 Definition: Vector, Vector Space What are vectors, and what are their coordinates? What is a vector space over the real field, origin, null vector, norm? 1.7.2 Theorem Suppose x, y, z ∈ R k Prove that and α is real. (a) |⃗x| ≥ 0; (b) |⃗x| = 0 if and only if ⃗x = 0; (c) |α⃗x| = |α||⃗x|; (d) |⃗x · ⃗y| ≤ |⃗x||⃗y|; (e) |⃗x + ⃗y| ≤ |⃗x| + |⃗y| (f) |⃗x − ⃗z| ≤ |⃗x − ⃗y| + |⃗y − ⃗z| 1.7.3 Remarks Theorem 1.0.37 (a), (b), and (f) allows us to regard R k as a metric space. (see Chap. 2). R 1 is called the line; R 2 is called the plane, or the complex plane. In these two cases the norm is just the absolute value of the corresponding real or complex number. 1.8 Appendix 1.8.1 Step 1 The members of R will be certain subsets of Q, called cuts. What three properties, by definition, does a cut (any set α) have? The letters p, q, r, will always denote rational numbers, and α, β, γ, will denote cuts. 1.8.2 Step 2 Establish R as an ordered set. 1.8.3 Step 3 Prove that the the ordered set R has the least-upper-bound property. 1.8.4 Step 4 If α, β ∈ R, define α + β to be the set of all sums r + s where r ∈ α, s ∈ β. Define 0* to be the set of all negative rational numbers. It is clear that 0* is a cut (see Step 1). Verify that the axioms for addition hold in R, with 0* playing the role of 0.
1.9. CONCEPTS FROM THE 2ED 11 1.8.5 Step 5 Having proved that the addition defined in Step 4 satisfies Axioms (A) of Definition 1.12, it follows that Proposition 1.14 is valid in R. Prove one of the requirements of Definition 1.17: If α, β, γ in R and β < γ then, α + β < α + γ and that α > 0* if and only if −α < 0*. 1.8.6 Step 6 Multiplication is a little more bothersome than addition in the present context, since products of negative rationals are positive. For this reason we confine ourselves first to R + , the set of all α ∈ R with α > 0*. If α, β ∈ R + , define αβ to be the set of all p such that p ≤ rs for some choice of r ∈ α, s ∈ β, r > 0, s > 0. Define 1* to be the set of all q < 1. Then, prove that the axioms (M) and (D) of Definition 1.12 hold, with R + in place of F and with 1* in the role of 1. 1.8.7 Step 7 Define αβ, and prove the distributive law α(β + γ) = αβ + αγ. Thus completing the proof that R is an ordered field with least-upper-bound property. 1.8.8 Step 8 Associate with each r ∈ Q the set r* which consists of all p ∈ Q such that p < r. It is clear that each r* is a cut; that is, r* ∈ R. Prove that these cuts satisfy the following relations: (a) r* + s* = (r + s)*; (b) r*s* = (rs)*; (c) r* < s* if and only if r < s 1.8.9 Step 9 1.9 Concepts From the 2ed Some these definitions and theorems are covered in the appendix in the 3rd edition (see above). 1.9.1 Definition What is a cut? 1.9.2 Theorem If p ∈ α, and q ∉ α, prove that p < q. 1.9.3 Theorem Let r be rational. Let α be the set consisting of all rational p such that p < r. Prove that α is a cut, and r is the smallest upper number of α. 1.9.4 Definition What is a rational cut? 1.9.5 Definition Let α and β be cuts. Define α = β and α ≠ β.
Page 1: Studying Rudin’s Principles of Ma
Page 4 and 5: iv CONTENTS 1.7 Euclidean Spaces .
Page 6 and 7: 2 CONTENTS
Page 8 and 9: 4 CHAPTER 1. THE REAL AND COMPLEX N
Page 16 and 17: 12 CHAPTER 1. THE REAL AND COMPLEX
Page 18: 14 CHAPTER 1. THE REAL AND COMPLEX

Studying Rudin's Principles of Mathematical Analysis Through ...

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?