## Elements of Abstract AnalysisIn nature's infinite book ofsecrecy A little I can read. Antony and Cleopatra, l. ii. This is a book about a few elementary concepts of analysis and the mathe matical structures which enfold them. It is more concerned with the interplay amongst these concepts than with their many applications. The book is self-contained; in the first chapter, after acknowledging the fundamental role ofmathematical logic, wepresent seven axioms of Set Theory; everything else is developed from these axioms. It would therefore be true, if misleading, to say that the reader requires no prior knowledge of mathematics. In reality, the reader we have in mind has that level of sophistication achieved in about three years of undergraduate study of mathematics and is already well acquainted with most of the structures discussed-rings, linear spaces, metric spaces, and soon-and with many ofthe principal analytical concepts convergence, connectedness, continuity,compactness and completeness. Indeed, it is only after gaining familiarity with these concepts and their applications that it is possible to appreciate their place within a broad framework of set based mathematics and to consolidate an understanding of them in such a framework. To aid in these pursuits, wepresent our reader with things familiar and things new side by side in most parts of the book-and we sometimes adopt an unusual perspective. That this is not an analysis textbook is clear from its many omissions. |

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Pagina 13

If A is any superset of field(r), then r may be described as a relation ON A. The relation r is said to be ONE-TO-ONE or

If A is any superset of field(r), then r may be described as a relation ON A. The relation r is said to be ONE-TO-ONE or

**INJECTIVE**if and only if, for each y in ran(r), there is precisely one v in dom(r) such that (x, y) € r. Pagina 14

If f is an

If f is an

**injective**function, then {(y,z) (x, y) e f} is a function; it is denoted by f" and called the INVERSE of f; this inverse is clearly**injective**also and has inverse f. If f is a function and X and Y are supersets of dom(f) and ... Pagina 16

This quotient map is surjective onto X/~ but is not, except in trivial cases,

This quotient map is surjective onto X/~ but is not, except in trivial cases,

**injective**. Example 1.2.15 Suppose that X is a set and that F is a collection of functions with domain X. For each a € X, the set {(f, f(x)) | f e P} is a ... Pagina 18

The range {vi e X i e I} of an

The range {vi e X i e I} of an

**injective**family w. I → X is called an INDEXED SUBSET of X; if the family is bijective, then the range, X itself, is called an INDEXED SET and we say that X is INDEXED by I. If the members of X are sets ... Pagina 19

It is clear that the restriction T; 2 of Tj to X, is

It is clear that the restriction T; 2 of Tj to X, is

**injective**; and, for each a € Xj, y = {(j, a)} U z\{(j, zj)} e Xj,” and Tj,2(y) = a, so that Tj,” (and hence also Tj) is surjective. Note that surjectivity depends on the non-emptiness ...### Ce spun oamenii - Scrie o recenzie

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### Cuprins

Counting | 41 |

Algebraic Structure | 57 |

Analytic Structure | 91 |

Linear Structure | 115 |

Geometric Structure | 133 |

Topological Structure | 159 |

Continuity and Openness | 177 |

Connectedness | 207 |

Convergence | 215 |

Compactness | 231 |

Completeness | 245 |

Solutions | 269 |

Bibliography | 285 |

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acc(A algebra arbitrary Axiom of Choice Banach bijective bounded called cardinal closed subset commutative compact space converges convex counting number defined Definition denoted dense disjoint dom(r domain endowed ensures equivalent Example EXERCISES Q exists f is continuous field F filter finite subset follows Hausdorff space Hilbert space homomorphism includes induction initial topology injective injective function intersection inverse isometric Lemma linear subspace linearly independent maximal subspace metric space nbd(x non-empty set non-empty subset non-trivial normed linear space notation oo U oo open ball open neighbourhood open subset order isomorphism ordered set ordinal Proof Let Proof Suppose ps(X quotient real numbers Recursive relative topology second countable semimetric space seminormed seminormed linear space Show spaces and f subbase Suppose f surjective topological space topology determined totally ordered ultrafilter union unique unit ball vector space whence