## 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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### Cuprins

List of Symbols xi | 1 |

I | 6 |

3 | 20 |

Counting | 41 |

Algebraic Structure | 57 |

Analytic Structure | 91 |

Linear Structure | 115 |

Geometric Structure | 133 |

Connectedness | 207 |

Convergence | 215 |

Compactness | 231 |

Completeness | 245 |

Solutions | 269 |

invS | 274 |

COS convex hull | 280 |

Bibliography | 285 |

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### Termeni și expresii frecvente

algebra arbitrary Axiom ball Banach space base basis bijective bounded called cardinal certainly Choice clearly closed collection compact complete condition connected continuous converges convex countable defined Definition denoted dense determined disjoint domain easy element empty endowed ensures equivalent Example exists fact filter finite follows function given Hausdorff homomorphism ideal identity implies includes independent induction injective intersection interval inverse isometric isomorphism limit linear space linearly maximal metric space Moreover multiplication natural neighbourhood non-empty set normed linear space open subset operations ordered ordered set ordinal otherwise particular Proof Suppose proper properties proved quotient relation respect result ring satisfies scalar semimetric separable sequence Show similar structure subspace Suppose X Theorem topological space topology totally ordered union unique unital usual vector space wedge whence yields