## 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 20

1.3 Ordered Sets Order Relations Definition 1.3.1 Suppose S is a set. ... ORDER RELATION on S. If ~ is such a relation on S, then the ordered pair (S, <) is called a

1.3 Ordered Sets Order Relations Definition 1.3.1 Suppose S is a set. ... ORDER RELATION on S. If ~ is such a relation on S, then the ordered pair (S, <) is called a

**PARTIALLY ORDERED**SET and we say that S is**partially ordered**by ~. Pagina 21

This book is concerned with sets endowed with various forms of structure, partial ordering being an example. ... But, not wishing to be pedantic, we shall refer to S itself as being a

This book is concerned with sets endowed with various forms of structure, partial ordering being an example. ... But, not wishing to be pedantic, we shall refer to S itself as being a

**partially ordered**set (or group, or vector space, ... Pagina 22

P(S) is

P(S) is

**partially ordered**by inclusion; provided S # 2, the order relation is not empty. P(S) has maximum element S and minimum element Ø. Example 1.3.7 Suppose X and S are sets and S is endowed with a partial ordering <. Pagina 23

WELL ORDERED by & if and only if each non-empty subset of S has a minimum element. (S, <), or simply S, is called a

WELL ORDERED by & if and only if each non-empty subset of S has a minimum element. (S, <), or simply S, is called a

**TOTALLY ORDERED**, DENSELY ORDERED, COMPLETELY ORDERED or WELL ORDERED set as may be appropriate. Pagina 24

Similarity Some bijective maps between sets endowed with a

Similarity Some bijective maps between sets endowed with a

**total ordering**preserve that ordering. Such maps are called similarity functions. Although there is no set of all**totally ordered**sets, similarity behaves like an equivalence ...### 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 |

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### 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