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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 ff erent weights or densities including singular weights. (d) Characterization of dual spaces. (e) Completeness. (2) In fi 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 : x / A } . We also de fi ne the symmetric di ff 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 fi 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 fi 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 fi nitely often and { A n a.a. } as A n almost al- ways.) Then x { A n i.o. } i ff 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 ff 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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2 BRUCE K. DRIVER 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 fi 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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