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 18
... set I will then be called the INDEXING SET of the family , the members of I ... ordered pairs , it is usual to leave out parentheses ; we write , for ... set X , the identity function x is a bijective family . So X can be indexed by ...
... set I will then be called the INDEXING SET of the family , the members of I ... ordered pairs , it is usual to leave out parentheses ; we write , for ... set X , the identity function x is a bijective family . So X can be indexed by ...
Pagina 20
... Ordered Sets Order Relations Definition 1.3.1 Suppose S is a set . A relation on S which is both anti - reflexive and transitive is called a PARTIAL ORDER RELATION on S. If < is such a relation on S , then the ordered pair ( S , < ) is ...
... Ordered Sets Order Relations Definition 1.3.1 Suppose S is a set . A relation on S which is both anti - reflexive and transitive is called a PARTIAL ORDER RELATION on S. If < is such a relation on S , then the ordered pair ( S , < ) is ...
Pagina 21
Mícheál O'Searcoid. This book is concerned with sets endowed with various forms of structure , partial ordering being an example . An ordered pair such as ( S , < ) is used in the definition in order to ensure that the object is a set ...
Mícheál O'Searcoid. This book is concerned with sets endowed with various forms of structure , partial ordering being an example . An ordered pair such as ( S , < ) is used in the definition in order to ensure that the object is a set ...
Pagina 22
... ordered by inclusion ; provided SØ , the order relation is not empty . P ( S ) has maximum element S and minimum ... set . S is said to be • TOTALLY ORDERED by < if and only if , for every pair of distinct members x , y Є S , either x ...
... ordered by inclusion ; provided SØ , the order relation is not empty . P ( S ) has maximum element S and minimum ... set . S is said to be • TOTALLY ORDERED by < if and only if , for every pair of distinct members x , y Є S , either x ...
Pagina 23
... ORDERED , DENSELY ORDERED , COM- PLETELY ORDERED or WELL ORDERED set as may be appropriate . It is an easy calculation that every well ordered set is totally ordered and , indeed , com- pletely ordered ( Q 1.3.3 ) . We shall call < a total ...
... ORDERED , DENSELY ORDERED , COM- PLETELY ORDERED or WELL ORDERED set as may be appropriate . It is an easy calculation that every well ordered set is totally ordered and , indeed , com- pletely ordered ( Q 1.3.3 ) . We shall call < a total ...
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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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