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 3
... denoted by ZF . It is based on an axiomatic theory of mathematical logic in which a few self - evident truths , called LOGICAL AXIOMS , are represented by finite sequences of symbols ; one infers from these , in a finite number of ...
... denoted by ZF . It is based on an axiomatic theory of mathematical logic in which a few self - evident truths , called LOGICAL AXIOMS , are represented by finite sequences of symbols ; one infers from these , in a finite number of ...
Pagina 5
... denoted by particular symbols ; but is the same true for objects which are not well defined ? Specifically , if there exists a set which satisfies a particular condition , is it valid to assign to some such object a previously unused ...
... denoted by particular symbols ; but is the same true for objects which are not well defined ? Specifically , if there exists a set which satisfies a particular condition , is it valid to assign to some such object a previously unused ...
Pagina 8
... denoted by bc when the intended superset is clear from the context . Theorem 1.1.4 There exists exactly one set which has no members ; it is a subset of every set . There is no set which has every set as a member . Proof We have already ...
... denoted by bc when the intended superset is clear from the context . Theorem 1.1.4 There exists exactly one set which has no members ; it is a subset of every set . There is no set which has every set as a member . Proof We have already ...
Pagina 9
... denoted by Ua and called the UNION of the members of a . Definition 1.1.8 Suppose a is a set . We define the INTERSECTION of the members of a , denoted by Пa , to be { x € Ua | Vz E a , x € z } , which is a set by Axiom IV and the ...
... denoted by Ua and called the UNION of the members of a . Definition 1.1.8 Suppose a is a set . We define the INTERSECTION of the members of a , denoted by Пa , to be { x € Ua | Vz E a , x € z } , which is a set by Axiom IV and the ...
Pagina 13
... denoted by field ( r ) , is defined to be UUr ; this is a set by Axiom IV . It has two important subsets , namely the DOMAIN { x Є field ( r ) | Jy : ( x , y ) E r } of r , denoted by dom ( r ) , and the RANGE { y Є field ( r ) | 3x ...
... denoted by field ( r ) , is defined to be UUr ; this is a set by Axiom IV . It has two important subsets , namely the DOMAIN { x Є field ( r ) | Jy : ( x , y ) E r } of r , denoted by dom ( r ) , and the RANGE { y Є field ( r ) | 3x ...
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