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Unformatted text preview: 6.1 Integrals of continuous functions 83 Definition 6.1.8. A norm on a vector space is a function from X to R that satisfies (i) k x k ≥ , k x k = 0 ⇐⇒ x = 0. (ii) k ax k =  a  · k x k , ∀ a ∈ R . (iii) k x z k ≤ k x y k + k y z k , ∀ x,y,z ∈ X . Example 6.1.1. On R , absolute value is a norm. For x = ( a,b ) ∈ R 2 , k x k = √ a 2 + b 2 . For f ∈ R ( D ), k f k 1 := R D  f ( x )  dx . Definition 6.1.9. A step function is one which is locally constant except for finitely many points. Thus, a step function is usually defined in terms of a partition; it is constant on each subinterval and can have a jump discontinuity at each point of the partition. Example 6.1.2. Define the step functions f P ( x ) = inf x i 1 ≤ x ≤ x i f ( x ) and f P ( x ) = sup x i 1 ≤ x ≤ x i f ( x ) . Then the previous theorem states that f is integrable iff there is a sequence of partitions { P n } with Z  f P n ( x ) f P n ( x )  dx n →∞→ ....
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This note was uploaded on 11/26/2011 for the course MAT 4944 taught by Professor Wong during the Fall '11 term at FSU.
 Fall '11
 Wong
 Integrals, Vector Space

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