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 4
... exists . We shall employ the following logical symbols : the UNIVERSAL QUANTIFIER V and the EXISTENTIAL QUANTIFIER which we shall interpret as meaning for all and there exists respectively ; 3 ! which will mean there exists exactly one ...
... exists . We shall employ the following logical symbols : the UNIVERSAL QUANTIFIER V and the EXISTENTIAL QUANTIFIER which we shall interpret as meaning for all and there exists respectively ; 3 ! which will mean there exists exactly one ...
Pagina 5
... exists in mathematical logic , and in mathematics generally , is intrinsically non - specific : there exists an object which satisfies ... is the negation of for every object , that object does not satisfy .... It is possible that there ...
... exists in mathematical logic , and in mathematics generally , is intrinsically non - specific : there exists an object which satisfies ... is the negation of for every object , that object does not satisfy .... It is possible that there ...
Pagina 6
... exists a set whose members are precisely those sets x which satisfy the condition that every member of x is a member ... exists a set b whose members are precisely those sets y for which there exists x Є a for which ø ( x , y ) holds ...
... exists a set whose members are precisely those sets x which satisfy the condition that every member of x is a member ... exists a set b whose members are precisely those sets y for which there exists x Є a for which ø ( x , y ) holds ...
Pagina 7
... exists . What the axiom states is that , if we restrict this ' function ' to a ' domain ' a which is known to be a set , then the ' range ' is also a set . This cannot be proved from the other axioms of ZFC , but it is not an ...
... exists . What the axiom states is that , if we restrict this ' function ' to a ' domain ' a which is known to be a set , then the ' range ' is also a set . This cannot be proved from the other axioms of ZFC , but it is not an ...
Pagina 8
... exists a set a to which every such a belongs ; where we adopt this loose convention , the reader is encouraged to make the mental check that some such set a does indeed exist . Sometimes several conditions will be indicated ; they might ...
... exists a set a to which every such a belongs ; where we adopt this loose convention , the reader is encouraged to make the mental check that some such set a does indeed exist . Sometimes several conditions will be indicated ; they might ...
Cuprins
sup | 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