Download e-book for iPad: An Introduction to Metric Spaces and Fixed Point Theory by Mohamed A. Khamsi

By Mohamed A. Khamsi

ISBN-10: 0471418250

ISBN-13: 9780471418252

Offers up to date Banach area results.
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Example text

Its limit will necessarily lie in K proving that K is compact. Note also that if an infinite number of terms of {xn} are the same then we could select a subsequence of {xn} which is constant and therefore converges trivially. So by throwing away some terms if necessary, we may assume that each two terms of {xn} are distinct. Since K is precompact there exist points {21,1,22,1, ·· · ,2 n ,,i} Q K such that each point of K lies within distance at most 1 from at least one of these points, that is, ~Kç{jB{ziy,\).

It is easily verified that (i) the above limit does indeed exist; (ii) the space {Mc,d") is complete; and (iii) M is isometric to the subspace of (Mc,d*) consisting of all equivalence classes of the form [{x}], x € M. We now look at some specific examples. 30 CHAPTER 2. METRIC SPACES Example 1. Completeness of the space R of real numbers (with usual distance d(x, y) — \x - y\) is a fundamental consequence of the sup axiom. Suppose {xn} is a Cauchy sequence of real numbers and define the two sequences {un} and {vn} as follows: un = sup{xfc : A; > n}; vn = inf{xfc : k > n).

6/ 36 CHAPTER 2. METRIC SPACES Unfortunately, for our purposes, there is even a problem with this abstract formulation. In the study of metric fixed point theory closed balls turn out to play a fundamental role. A subset A of a bounded metric space M will be said to be an admissible subset of M if A can be written as the intersection of a family of closed balls centered at points of M. The family A{M) of all admissible subsets of M enters into the study of metric fixed point theory in a very natural way and is therefore the obvious candidate for the needed underlying convexity structure.

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An Introduction to Metric Spaces and Fixed Point Theory by Mohamed A. Khamsi


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