New PDF release: An Introduction to Abstract Analysis

By W. A. Light (auth.)

ISBN-10: 0412310902

ISBN-13: 9780412310904

ISBN-10: 1489972544

ISBN-13: 9781489972545

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Extra resources for An Introduction to Abstract Analysis

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Hence {xk(i)} is a Cauchy sequence of real numbers for each 1 :::; i :::; n, and so converges to a limit, which we call x( i). In this way we define the element x in 1Rn, and as xk( i) -+ x( i), 1 ~ i ~ n, we have Xk -+ x 0 • Hence any Cauchy • sequence in 1Rn is convergent to a point in 1Rn, as required. We will conclude this section with an important theorem known as Cantor's intersection theorem. This result will be used in later chapters, but belongs naturally in this section on completeness.

Both selections violate the stipulation that the selection be increasing. 12 If {xn} is a Cauchy sequence in a normed linear space X, and some subsequence {xnk} converges to x in X, then {xn} itself converges to x. Proof. We start in the usual manner. Take f > 0. Since {xn} is Cauchy there is an N such that llxp- x 9 11 ~ whenever p, q ~ N. Since {Xnk} converges to x, there is an m such that nm ~ N and llxnm - xll ~ Now applying the first statement with q = nm gives llxp - Xnm II ~ ~f for all p ~ N.

It is easy to manufacture dense sets for a. s small and as simple as possible. 2 Let X be a normed linear space. statements are equivalent: (i) the set A is dense in X, (ii} the set X \ A has no interior point. Then the following Proof. ( i) =? ( ii) We use the technique of contrapositive argument here. We show that if X \ A has an interior point then A ::j:. X. If x is an interior point of X \A, then there is a ball Br (x) with r > 0 which is contained completely in X \ A. It is impossible for this point x to lie in A and so A ::j:.

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An Introduction to Abstract Analysis by W. A. Light (auth.)

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