# Download e-book for kindle: Aspects of mathematics and its applications, no TOC and by Jorge Alberto Barroso

By Jorge Alberto Barroso

ISBN-10: 0444877274

ISBN-13: 9780444877277

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Extra info for Aspects of mathematics and its applications, no TOC and Index

Sample text

If the space R is not complete, it is always possible to embed it in an entirely definite manner in a complete space. DEFINITION 2. Let R be an arbitrary metric space. A complete metric space R* is said to be the completion of the space R if: 1) R is a subspace of the space R*; and 2) R. e. [R] = R*. ) For example, the space of all real numbers is the completion of the space of rationals. -to ~IETRW RPACES [CB. II TH1':OHl::~1 2. J~'l'l'/"y /net ric space has a completion and all of its completions arc isolllctric.

There is a one-to-one mapping", of the f'pace R* onto R** such that 1) ",(x) = x for all J' (: R; and 2) if :r* ~ x** and y* ~ y**, then p(x*, y*) = p(x**, y**). Sueh a mappillg '" is defilled in the following way. Let x* be an arbitrary point of R*. ;r" I of points in R which converges to x*. But the sequence IXn) can be assumed to belong also to R**. Since R** is complete, Ix,,) converges in R** to some point :c**. r*) = x**. It is dear that this correspondence is one-to-one and docs not depend on the choice of the sequence Ix n ) which converges to the point x*.

Theorems 2 and 3 carryover automatically to continuous mappings of topological spaces. §13. e. the fact that every fundamental sequence of real numbers converges to some limit. The real line represents the simplest example. of the so-called complete metric spaces whose basic properties we shall consider in this section. e. rists an N. , n" ~ N •. It follows directly from the triangle axiom that eyery convergent sequence is fundamental. t',,} com'erges to x, then for giyen e > 0 it is possible to find a natural number N.