## 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. |

### Din interiorul cărții

Rezultatele 1 - 5 din 32

Pagina viii

It is an awesome thought that, through the achievements of modern mathematics—with its identification of important concepts and its highly developed language and concise

It is an awesome thought that, through the achievements of modern mathematics—with its identification of important concepts and its highly developed language and concise

**notation**-quite ordinary mathematicians can see further and delve ... Pagina 2

We shall introduce some

We shall introduce some

**notation**, give some formal definitions and establish some conventions; and we shall record some basic results which will subsequently be used freely. We shall see that Set Theory contains some perhaps unexpected ... Pagina 4

**Notation**Even in intuitive set theory, members of sets may themselves be sets, so that a particular object may be regarded as a set or as a member, depending on the context. In ZFC, and within the mathematics that emerges from it, ... Pagina 6

... of declaring a set by listing its members in between braces; so the set referred to above can be presented as {a, b} or as {b, a); if a = b then it will more usually be presented as {a} or as {b}.

... of declaring a set by listing its members in between braces; so the set referred to above can be presented as {a, b} or as {b, a); if a = b then it will more usually be presented as {a} or as {b}.

**Notation**sometimes hides pitfalls. Pagina 8

If a is a set and p(x) is a condition on sets, then we shall adopt the

If a is a set and p(x) is a condition on sets, then we shall adopt the

**notation**{a, C a | p(x)} as an abbreviation for {a e P(a) | p(x)}; it requires both Axiom II and the Subset Principle to establish that this is a set.### Ce spun oamenii - Scrie o recenzie

Nu am găsit nicio recenzie în locurile obișnuite.

### 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 |

### Alte ediții - Afișează-le pe toate

### Termeni și expresii frecvente

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