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 viii
... defined or is to remain undefined in the book . Most of the notation we use is either standard or has been used by other authors ; but the reader will find a few innovations . My thanks are due to the SUMS advisors for their helpful ...
... defined or is to remain undefined in the book . Most of the notation we use is either standard or has been used by other authors ; but the reader will find a few innovations . My thanks are due to the SUMS advisors for their helpful ...
Pagina x
... Defined by Functions . 177 187 8.3 Derived Topological Spaces 191 8.4 Topologies on Linear Spaces . 202 9 . Connectedness . ..207 9.1 Connected Spaces . 207 9.2 Pathwise Connectedness 10. Convergence . 10.1 Filters . 10.2 Limits . 11 ...
... Defined by Functions . 177 187 8.3 Derived Topological Spaces 191 8.4 Topologies on Linear Spaces . 202 9 . Connectedness . ..207 9.1 Connected Spaces . 207 9.2 Pathwise Connectedness 10. Convergence . 10.1 Filters . 10.2 Limits . 11 ...
Pagina 2
... defined property which is formalizable in a sense determined by the underlying logical theory , we cannot avoid the difficulty demonstrated by Bertrand Russell's famous question : does the class of all those sets which are not members ...
... defined property which is formalizable in a sense determined by the underlying logical theory , we cannot avoid the difficulty demonstrated by Bertrand Russell's famous question : does the class of all those sets which are not members ...
Pagina 3
Mícheál O'Searcoid. to an accumulation of objects determined by some well defined and formalizable property we shall call it a CLASS ; such a class may or may not be a set . A MEMBER of a set will variously be called an ELEMENT or a ...
Mícheál O'Searcoid. to an accumulation of objects determined by some well defined and formalizable property we shall call it a CLASS ; such a class may or may not be a set . A MEMBER of a set will variously be called an ELEMENT or a ...
Pagina 5
... defined objects of a logical theory can be 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 ...
... defined objects of a logical theory can be 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 ...
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