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Unformatted text preview: Math 528 Jan 11, 2005 1 Geometry and Topology II Fall 2005, PSU Lecture Notes 1 1 Topological Manifolds The basic objects of study in this class are manifolds. Roughly speaking, these are objects which locally resemble a Euclidean space. In this section we develop the formal definition of manifolds and construct many examples. 1.1 The Euclidean space By R we shall always mean the set of real numbers, which is a well ordered field with the following property. Completeness Axiom . Every nonempty subset of R which is bounded above has a least upper bound . The set of all ntuples of real numbers R n := { ( p 1 , . . . , p n )  p i ∈ R } is called the Euclidean nspace. So we have p ∈ R n ⇐⇒ p = ( p 1 , . . . , p n ) , p i ∈ R . Let p and q be a pair of points (or vectors) in R n . We define p + q := ( p 1 + q 1 , . . . , p n + q n ). Further, for any scalar r ∈ R , we define rp := ( rp 1 , . . . , rp n ). It is easy to show that the operations of addition and scalar multiplication that we have defined turn R n into a vector space over the field of real numbers. Next we define the standard inner product on R n by p, q = p 1 q 1 + . . . + p n q n . Note that the mapping · , · : R n × R n → R is linear in each variable and is sym metric. The standard inner product induces a norm on R n defined by p := p, p 1 / 2 . If p ∈ R , we usually write  p  instead of p . Theorem 1.1.1. (The CauchySchwartz inequality) For all p and q in R n  p, q  p q . The equality holds if and only if p = λq for some λ ∈ R . 1 Last revised: October 2, 2006 1 Proof I. If p = λq it is clear that equality holds. Otherwise, let f ( λ ) := p λq, p λq . Then f ( λ ) > 0. Further, note that f ( λ ) may be written as a quadratic equation in λ : f ( λ ) = p 2 2 λ p, q + λ 2 q 2 . Hence its discriminant must be negative: 4 p, q 2 4 p 2 q 2 < which completes the proof. Proof II. Again suppose that p = λq . Then p, q = p q p p , q q . Thus it suffices to prove that for all unit vectors p and q we have  p, q  ≤ 1 , and equality holds if and only if p = ± q . This may be proved by using the method of lagrangne multipliers to find the maximum of the function x, y subject to the constraints x = 1 and y = 1. More explicitly we need to find the critical points of f ( x, y, λ 1 , λ 2 ) := x, y + λ 1 ( x 2 1) + λ 2 ( y 2 1) = n i =1 ( x i y i + λ 1 x 2 i + λ 2 y 2 i ) λ 1 λ 2 . At a critical point we must have 0 = ∂f/∂x i = y i + 2 λ 1 x i , which yields that y = ± x . The standard Euclidean distance in R n is given by dist( p, q ) := p q . The proof of the following fact is left as an exercise. Corollary 1.1.2. (The triangle inequality) For all p , q , r in R n ....
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This note was uploaded on 08/25/2011 for the course MATH 6456 taught by Professor Staff during the Spring '08 term at Georgia Institute of Technology.
 Spring '08
 Staff
 Logic, Geometry, Topology

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