Lecture Notes - Mth 511

Lecture Notes - Mth 511 - MTH 511 - REAL ANALYSIS I OREGON...

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Unformatted text preview: MTH 511 - REAL ANALYSIS I OREGON STATE UNIVERSITY FALL 2011 TEXT: REAL ANALYSIS, N.L. Carothers L ECTURE N OTES Chapter 3 Metrics and Norms Let M be a set and let M × M = { ( x,y ) : x,y ∈ M } . Definition: A function d : M × M → R is a metric on M if and only if each of the following hold (i) d ( x,y ) ≥ , for all x,y ∈ M . (ii) d ( x,y ) = 0 if and only if x = y . (iii) d ( x,y ) = d ( y,x ) for all x,y ∈ M . (iv) d ( x,z ) ≤ d ( x,y ) + d ( y,z ) for all x,y,z ∈ M . This is known as the triangle inequality. Definition: A metric space is any pair ( M,d ) , where M is a set and d is a metric on M . Example: M = R , d ( x,y ) = | x- y | for all x,y ∈ R . Then ( R ,d ) is a metric space. Relation to Limits Suppose f : R → R and a ∈ R . Recall that lim x → a f ( x ) = L if and only if ∀ ε > , ∃ δ > such that | f ( x )- L | < ε whenever < | x- a | < δ . In terms of metrics, this becomes: lim x → a f ( x ) = L if and only if ∀ ε > , ∃ δ > such that d ( f ( x ) ,L ) < ε whenever < d ( x,a ) < δ . Or, even more generally, for f : M 1 → M 2 , and a ∈ M 1 , we have lim x → a f ( x ) = L if and only if ∀ ε > , ∃ δ > such that d 2 ( f ( x ) ,L ) < ε whenever < d 1 ( x,a ) < δ. Normed Vector Spaces Let V be a vector space over R (or C ). Definition: Then k · k : V → R is a norm on V if and only if each of the following hold (i) k x k ≥ , ∀ x ∈ V . (ii) k x k = 0 if and only if x = 0 , the additive identity in V . (iii) k αx k = | α |k x k , ∀ α ∈ R (or C ), ∀ x ∈ V . 1 (iv) k x + y k ≤ k x k + k y k , ∀ x,y ∈ V Note that a norm induces a metric by d ( x,y ) = k x- y k . This clearly satisfies each of the first three properties of a metric, and it satisfies the triangle inequality since k x- z k = k ( x- y ) + ( y- z ) k ≤ k x- y k + k y- z k . Examples: Let V = R n := { ( x 1 ,x 2 ,...,x n ) : x i ∈ R } . Then the following are norms on R n : 1. k x k 1 := ∑ n i =1 | x i | , ∀ x ∈ R n . 2. k x k 2 := ( ∑ n i =1 | x i | 2 ) 1 / 2 , ∀ x ∈ R n . 3. k x k ∞ := max i ≤ n | x i | . 4. k x k p := ( ∑ n i =1 | x i | p ) 1 /p , for 1 ≤ p < ∞ . Exercise: Prove that lim p →∞ k x k p = k x k ∞ . (Factor out highest abs., take limit of what remains). Corresponding Metrics: k x- y k 1 = X | x i- y i | k x- y k 2 = X | x i- y i | 2 1 / 2 k x- y k ∞ = max | x i- y i | More Examples: Let ( x n ) be a sequence in R , and let C [ a,b ] denote the set of continuous real-valued functions on the interval [ a,b ] . Note that C [ a,b ] is a vector space over R . 1. Define l 1 := { ( x n ) ⊂ R : ∑ ∞ n =1 | x n | < ∞} . Then ( l 1 , k · k 1 ) is a normed vector space....
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This note was uploaded on 11/10/2011 for the course MATH 511 taught by Professor Higdon during the Spring '11 term at Oregon State.

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Lecture Notes - Mth 511 - MTH 511 - REAL ANALYSIS I OREGON...

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