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

By Mohamed A. Khamsi

ISBN-10: 0471418250

ISBN-13: 9780471418252

ISBN-10: 1118033078

ISBN-13: 9781118033074

Content material:
Chapter 1 advent (pages 1–11):
Chapter 2 Metric areas (pages 13–40):
Chapter three Metric Contraction ideas (pages 41–69):
Chapter four Hyperconvex areas (pages 71–99):
Chapter five “Normal” buildings in Metric areas (pages 101–124):
Chapter 6 Banach areas: creation (pages 125–170):
Chapter 7 non-stop Mappings in Banach areas (pages 171–196):
Chapter eight Metric fastened aspect thought (pages 197–241):
Chapter nine Banach area Ultrapowers (pages 243–271):

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Extra info for An Introduction to Metric Spaces and Fixed Point Theory

Sample text

27 COMPLETENESS The diameter of a nonempty bounded subset D of a metric space is the number diam (D) = sup{d(x, y) : x,y 6 D). Note that we have diam (D) = diam (£>). 10 (Cantor's Intersection Theorem) A metric space (M,d) is complete if and only if given any descending sequence {Dn} of nonempty bounded closed subsets of M, Um diam (D„) = 0 => n—»oo I ' n=l Γ\ΌηφΖ. Proof. Suppose M is complete and let {Dn} be a descending sequence of nonempty bounded closed subsets of M for which lim diam(D n ) = 0.

For any mapping F : M —> M we use k(F) to denote the Lipschitz constant of F; thus Note in particular that if F, G : M —> M are two lipschitzian mappings, then k{FoG) n, then m-l d{xn,Xm) < Σ, d x ( i>xi+l)- i=n oo Consequently {x n } is a Cauchy sequence if Σ d{xi>xi+i) < °°> a n d since i=l i+l d(xi,xi+1) = d(T{x),T (x)) oo oo i=l i=l < k{T)d(x,T{x)) it follows that , Xi+l) < OO.

G Dk it follows that lim xnir = z. fc—»oo ■ The following general fact about completeness also is quite useful. We omit the very easy proof. 1 Every closed subspace of a complete metric space is itself complete. Here is another interesting fact about completeness. , S — M). Any metric space M is isometric with a dense subset of a complete metric space called the completion of M. ) One way to see this is to consider the space Mc of equivalence classes of all Cauchy sequences in M, where two Cauchy sequences {xn} and {zn} are said to be equivalent (written {xn} ~ {zn}) if hrn d(xn, zn) = 0.

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