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

Sample text

Proof. 10 is that statements (i) and (ii) are equivalent . We will now show that (i) and (iii) are equivalent. (iii) Let G be an open set in Y. It must be shown that f- 1 (G) is open in X. ) which also lies in f- 1 (G). Set Yo = f(x 0 ). , {y E Y : IIY - Yo II < r} ~ G. Since f is continuous, there exists a 8 > 0 such that whenever llx- xoll < 8, then IIJ(x)- J(xo)ll < r. Now the ball B8(xo) is our candidate for an open ball surrounding x 0 and lying in f- 1 (G). That this is indeed the case follows from the observation that f(B8(xo)) ~ Br(Yo) ~G.

Thus if we choose 6 = r/2 it is impossible to find an a in [0, 1] with lx- ai < 6, and sox is not a point of closure of [0, 1]. The only restriction we placed on x was that x should be greater than 1. Thus we may conclude that no point x with x > 1 is a point of closure of [0, 1]. e. [0, 1] is closed. Exercises 1. Let R have the usual norm. Which of the following sets are (a) open? (b) closed? {2}, [2,3], {t: t 2 < 1}, {2,3}, {t: 0 < t::::; 4}. 2. Show that int{intA) = intA and int{A n B) Give an example to show that int(A U B) necessarily true.

Note that this definition extends our existing notion that the quantity llx- all measures the distance between the points x and a in the normed linear space. The present definition says that the distance from a point x in X to a set A is the infimum of the individual distances of x from points in A. 12 need not be attained. When such a point a* does exist it is called a closest point or best approximation to x from A. e. the norm of a point is equal to its modulus) and A= (0, 1), then dist(2, A)= 1, but there is no point a* in (0, 1) such that llx- a*ll = 1.