Elements of Abstract AnalysisSpringer Science & Business Media, 6 dec. 2012 - 300 pagini In 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 iii
... Number Theory G.A. Jones and J.M. Jones Elements of Logic via Numbers and Sets D.L. Johnson Further Linear Algebra ... Real Analysis J.M. Howie Sets , Logic and Categories P. Cameron Symmetries D.L. Johnson Topics in Group Theory G ...
... Number Theory G.A. Jones and J.M. Jones Elements of Logic via Numbers and Sets D.L. Johnson Further Linear Algebra ... Real Analysis J.M. Howie Sets , Logic and Categories P. Cameron Symmetries D.L. Johnson Topics in Group Theory G ...
Pagina ix
... Numbers . Cardinality Enumeration 2.4 Cardinality of Unions and Products 3. Algebraic Structure .. Elementary ... Number Systems . 4.3 4.4 Real and Complex Functions Inequalities Ordered Algebraic Structure བྲ ཕྲག སྐྱབ 91 91 96 102 112 5 ...
... Numbers . Cardinality Enumeration 2.4 Cardinality of Unions and Products 3. Algebraic Structure .. Elementary ... Number Systems . 4.3 4.4 Real and Complex Functions Inequalities Ordered Algebraic Structure བྲ ཕྲག སྐྱབ 91 91 96 102 112 5 ...
Pagina xi
... numbers 97 set intersection 9 extended rational nos . . 100 set union .... 9 real numbers . .97 set complement . 8 R extended real nos . . 100 set intersection 10 T unit circle . . 102 set union . 10 Z integers .. ... 97 set difference ...
... numbers 97 set intersection 9 extended rational nos . . 100 set union .... 9 real numbers . .97 set complement . 8 R extended real nos . . 100 set intersection 10 T unit circle . . 102 set union . 10 Z integers .. ... 97 set difference ...
Pagina 4
... real numbers and so on , are presented as sets . It is therefore unnecessary to provide different axioms for objects which might otherwise be perceived as being of different types . Moreover , it is tacitly assumed when basing set ...
... real numbers and so on , are presented as sets . It is therefore unnecessary to provide different axioms for objects which might otherwise be perceived as being of different types . Moreover , it is tacitly assumed when basing set ...
Pagina 11
... numbers can be gathered together in a set ; but it also ensures , perhaps surprisingly , that real numbers can be presented as sets and that they too form a set . It thus opens the way for the development of mathematical analysis ...
... numbers can be gathered together in a set ; but it also ensures , perhaps surprisingly , that real numbers can be presented as sets and that they too form a set . It thus opens the way for the development of mathematical analysis ...
Cuprins
21 | |
Alls Well that Ends Well Viii | 29 |
Counting | 61 |
Algebraic Structure | 80 |
Analytic Structure | 91 |
Linear Structure | 115 |
Geometric Structure | 133 |
Topological Structure | 159 |
Continuity and Openness | 177 |
Connectedness | 207 |
Convergence | 215 |
Compactness | 231 |
91 | 242 |
Completeness | 245 |
Solutions | 269 |
Bibliography | 285 |
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Termeni și expresii frecvente
acc(A arbitrary Axiom of Choice bijective bounded called cardinal closed subset compact space compact subset complete connected converges Corollary counting number defined Definition denote dense disjoint domain endowed ensures equivalent Example EXERCISES Q exists field F filter finite subset follows ƒ is continuous Hausdorff space Hilbert space homomorphism includes induced inequality initial topology injective injective function inner product inverse Lemma linear subspace linearly independent maximal subspace maximal wedge metric space nbd(x non-empty set non-empty subset non-trivial normed linear space open ball open intervals open neighbourhood open sets open subset ordered set ordinal Proof Suppose ps(X ran(u real linear space Recursive relative topology second countable semimetric space seminormed seminormed linear space sequence sequentially Show subbase surjective T₁ topological space topology determined totally ordered ultrafilter union unique unit ball usual topology vector space whence