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 2
... show how relations and functions can be presented within the theory . We shall introduce some notation , give some formal definitions and establish some conventions ; and we shall record some basic results which will subsequently be ...
... show how relations and functions can be presented within the theory . We shall introduce some notation , give some formal definitions and establish some conventions ; and we shall record some basic results which will subsequently be ...
Pagina 10
... show that { { a } , { b , c } } is a set ; and { a , b , c } is a set by Axiom IV . Then { a } and { b , c } are subsets of { a , b , c } and { { a } , { b , c } } is a partition of { a , b , c } . Definition 1.1.12 Suppose a is a set ...
... show that { { a } , { b , c } } is a set ; and { a , b , c } is a set by Axiom IV . Then { a } and { b , c } are subsets of { a , b , c } and { { a } , { b , c } } is a partition of { a , b , c } . Definition 1.1.12 Suppose a is a set ...
Pagina 11
... Show that there exists a set { a , b , c , d } whose members are precisely a , b , c and d . Q1.1.2 Suppose a and b are sets and b ‡ Ø . Prove a \ Ub = { a \ y | y € b } . 1.2 Relations and Functions Sets are unordered ; this is 1. Sets 11.
... Show that there exists a set { a , b , c , d } whose members are precisely a , b , c and d . Q1.1.2 Suppose a and b are sets and b ‡ Ø . Prove a \ Ub = { a \ y | y € b } . 1.2 Relations and Functions Sets are unordered ; this is 1. Sets 11.
Pagina 19
... show that f1 ( UV ) = Uƒ ̃1 ( V ) and that f1 ( v ) = [ ƒ ̃1 ( V ) . Now suppose that UC P ( X ) and show that f ( UU ) Uƒ ( U ) and that f ( nu ) ≤nƒ ( U ) , but that this last inclusion may be proper ; find a condition on ƒ under ...
... show that f1 ( UV ) = Uƒ ̃1 ( V ) and that f1 ( v ) = [ ƒ ̃1 ( V ) . Now suppose that UC P ( X ) and show that f ( UU ) Uƒ ( U ) and that f ( nu ) ≤nƒ ( U ) , but that this last inclusion may be proper ; find a condition on ƒ under ...
Pagina 20
... Show that A ≤ ƒ ̃1 ( ƒ ( A ) ) with equality if ƒ is injective . Q 1.2.3 Suppose X and Y are sets , BC Y and f : X → Y. Show that ƒ ‹ ƒ ̃1 ( B ) ) = Bnf ( X ) . Deduce that f ( f1 ( B ) ) = B if ƒ is surjective . Q 1.2.4 Suppose ƒ : Y ...
... Show that A ≤ ƒ ̃1 ( ƒ ( A ) ) with equality if ƒ is injective . Q 1.2.3 Suppose X and Y are sets , BC Y and f : X → Y. Show that ƒ ‹ ƒ ̃1 ( B ) ) = Bnf ( X ) . Deduce that f ( f1 ( B ) ) = B if ƒ is surjective . Q 1.2.4 Suppose ƒ : Y ...
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