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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Rezultatele 6 - 10 din 83
Pagina 9
... exists a set whose members are precisely those sets y for which y = a or y b . It is unique by Axiom I. Example 1.1.6 Suppose a is a set . Apply the Pairing Principle with b = a . The result is that there exists a set whose only member ...
... exists a set whose members are precisely those sets y for which y = a or y b . It is unique by Axiom I. Example 1.1.6 Suppose a is a set . Apply the Pairing Principle with b = a . The result is that there exists a set whose only member ...
Pagina 10
... exists z Є b such that x z . So x Є a \ z , whence x € U { a \ y | y € b } . It follows that abcU { a \ y | y € b } . Conversely , suppose x E U { a \ y | y Є b } ; then there exists z Є b such that x Є a \ z . So x Є a and x ‡ z ...
... exists z Є b such that x z . So x Є a \ z , whence x € U { a \ y | y € b } . It follows that abcU { a \ y | y € b } . Conversely , suppose x E U { a \ y | y Є b } ; then there exists z Є b such that x Є a \ z . So x Є a and x ‡ z ...
Pagina 11
... exists . The axiom is designed to ensure that all natural 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 ...
... exists . The axiom is designed to ensure that all natural 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 ...
Pagina 17
... exists a bijective function between the two sets . In this case , the sets are said to be in one - to - one correspondence with each other . Although we might like to use this property to define an equivalence relation amongst sets , we ...
... exists a bijective function between the two sets . In this case , the sets are said to be in one - to - one correspondence with each other . Although we might like to use this property to define an equivalence relation amongst sets , we ...
Pagina 21
... exists . We say that A is ORDER - BOUNDED in S if and only if A is both bounded above and bounded below in S. • If the set of upper bounds for A in S has a minimal element , this is called a LEAST UPPER BOUND for A in S ; if there is ...
... exists . We say that A is ORDER - BOUNDED in S if and only if A is both bounded above and bounded below in S. • If the set of upper bounds for A in S has a minimal element , this is called a LEAST UPPER BOUND for A in S ; if there is ...
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