chap2-3

chap2-3 - A NALYSIS TOOLS W ITH APPLICATIONS 1 1....

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ANALYSIS TOOLS WITH APPLICATIONS 1 1. Introduction Not written as of yet. Topics to mention. (1) A better and more general integral. (a) Convergence Theorems (b) Integration over diverse collection of sets. (See probability theory.) (c) Integration relative to di f erent weights or densities including singular weights. (d) Characterization of dual spaces. (e) Completeness. (2) In f nite dimensional Linear algebra. (3) ODE and PDE. (4) Harmonic and Fourier Analysis. (5) Probability Theory 2. Limits, sums, and other basics 2.1. Set Operations. Suppose that X is a set. Let P ( X ) or 2 X denote the power set of X, that is elements of P ( X )=2 X are subsets of A. For A 2 X let A c = X \ A = { x X : x/ A } and more generally if A, B X let B \ A = { x B : A } . We also de f ne the symmetric di f erence of A and B by A 4 B =( B \ A ) ( A \ B ) . As usual if { A α } α I is an indexed collection of subsets of X we de f ne the union and the intersection of this collection by α I A α := { x X : α I 3 x A α } and α I A α := { x X : x A α α I } . Notation 2.1. We will also write ` α I A α for α I A α in the case that { A α } α I are pairwise disjoint, i.e. A α A β = if α 6 = β. Notice that is closely related to and is closely related to . For example let { A n } n =1 be a sequence of subsets from X and de f ne { A n i.o. } := { x X :# { n : x A n } = } and { A n a.a. } := { x X : x A n for all n su ciently large } . (One should read { A n i.o. } as A n in f nitely often and { A n a.a. } as A n almost al- ways.) Then x { A n i.o. } i f N N n N 3 x A n which may be written as { A n i.o. } = N =1 n N A n . Similarly, x { A n a.a. } i f N N 3 n N, x A n which may be written as { A n a.a. } = N =1 n N A n .
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2B R U C E K . D R I V E R 2.2. Limits, Limsups, and Liminfs. Notation 2.2. The Extended real numbers is the set ¯ R := R } , i.e. it is R with two new points called and −∞ . We use the following conventions, ± · 0=0 , ± + a = ± for any a R , + = and −∞ − ∞ = −∞ while ∞−∞ is not de f ned. If Λ ¯ R we will let sup Λ and inf Λ denote the least upper bound and greatest lower bound of Λ respectively. We will also use the following convention, if Λ = , then sup = −∞ and inf =+ . Notation 2.3. Suppose that { x n } n =1 ¯ R is a sequence of numbers. Then lim inf n →∞ x n = lim n →∞ inf { x k : k n } and (2.1) lim sup n →∞ x n = lim n →∞ sup { x k : k n } . (2.2) We will also write lim for lim inf and lim for lim sup . Remark 2.4 . Notice that if a k := inf { x k : k n } and b k := sup { x k : k n } , then { a k } is an increasing sequence while { b k } is a decreasing sequence. Therefore the limits in Eq. (2.1) and Eq. (2.2) always exist and lim inf n →∞ x n =sup n inf { x k : k n } and lim sup n →∞ x n =inf n sup { x k : k n } .
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This note was uploaded on 10/11/2010 for the course MATH 11 taught by Professor Smith during the Three '10 term at ADFA.

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chap2-3 - A NALYSIS TOOLS W ITH APPLICATIONS 1 1....

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