Intro to Analysis in-class_Part_39

Intro to Analysis in-class_Part_39 - 6.1 Integrals of...

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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 ≥ 0 , 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 →∞ -----→ 0 . Since we can always take a common refinement of two partitions, this amounts to saying that f is integrable iff it can be approximated by step functions, i.e.,
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