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Dieudonne J. — Foundation of Modern Analysis
Dieudonne J. — Foundation of Modern Analysis



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Название: Foundation of Modern Analysis

Автор: Dieudonne J.

Аннотация:

FOUNDATIONS OFMODERN ANALYSISEnlarged and Corrected PrintingJ. DIEUDONNEThis book is the first volume of a treatise which will eventually consist offour volumes. It is also an enlarged and corrected printing, essentiallywithout changes, of my Foundations of Modern Analysis, published in1960. Many readers, colleagues, and friends have urged me to write a sequelto that book, and in the end I became convinced that there was a place fora survey of modern analysis, somewhere between the minimum tool kitof an elementary nature which I had intended to write, and specialistmonographs leading to the frontiers of research. My experience of teachinghas also persuaded me that the mathematical apprentice, after taking the firststep of Foundations, needs further guidance and a kind of general birdseyeview of his subject before he is launched onto the ocean of mathematicalliterature or set on the narrow path of his own topic of research.Thus I have finally been led to attempt to write an equivalent, for themathematicians of 1970, of what the Cours dAnalyse of Jordan, Picard,and Goursat were for mathematical students between 1880 and 1920.It is manifestly out of the question to attempt encyclopedic coverage, andcertainly superfluous to rewrite the works of N. Bourbaki. I have thereforebeen obliged to cut ruthlessly in order to keep within limits comparable tothose of the classical treatises. I have opted for breadth rather than depth, inthe opinion that it is better to show the reader rudiments of many branchesof modern analysis rather than to provide him with a complete and detailedexposition of a small number of topics.Experience seems to show that the student usually finds a new theorydifficult tograsp at a first reading. He needs to return to it several times beforehe becomes really familiar with it and can distinguish for himself whichare the essential ideas and which results are of minor importance, and onlythen will he be able to apply it intelligently. The chapters of this treatise arevi PREFACE TO THE ENLARGED AND CORRECTED PRINTINGtherefore samples rather than complete theories: indeed, I have systematically tried not to be exhaustive. The works quoted in the bibliography willalways enable the reader to go deeper into any particular theory.However, I have refused to distort the main ideas of analysis by presentingthem in too specialized a form, and thereby obscuring their power andgenerality. It gives a false impression, for example, if differential geometryis restricted to two or three dimensions, or if integration is restricted to Lebesgue measure, on the pretext of making these subjects more accessible orintuitive.On the other hand I do not believe that the essential content of the ideasinvolved is lost, in a first study, by restricting attention to separable metrizabletopological spaces. The mathematicians of my own generation were certainlyright to banish, hypotheses of countability wherever they were not needed: thiswas the only way to get a clear understanding.


Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1969

Количество страниц: 387

Добавлена в каталог: 09.12.2008

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
Partial mapping      1.5
Partial sum (nth) of a series      5.2
Partition of a set      1.8
Path      9.6 and 10.2 prob.
Path reduced to a point      9.6
Peano curve      4.2 prob. and prob.
Peano’s existence theorem      10.5 prob.
Phragmen — Lindeloef’s principle      9.5 prob.
Picard’s Theorem      10.3 prob.
Piecewise linear function      8.7
Point      3.4
Pole of an analytic function      9.15
Positive definite Hermitian form      6.2
Positive Hermitian form      6.2
Positive Hermitian operator      11.5
Positive number      2.2
Power series      9.1
Precompact set      3.17
Precompact space      3.16
Prehilbert space      6.2
Primary factor      9.12 prob.
primitive      8.7
Principle of analytic continuation      9.4
Principle of extension of identities      3.15
Principle of extension of inequalities      3.15
Principle of isolated zeros      9.1
Principle of maximum      9.5
Product of a family of sets      1.8
Product of metric spaces      3.20
Product of normed spaces      5.4
Projection (first, second, zth)      1.3
Projections in a direct sum      5.4
Purely imaginary number      4.4
Pythagoras’s theorem      6.2
Quasi-derivative, quasi-differentiable function      8.4 prob.
Quasi-Hermitian operator      11.5 prob.
Quotient set      1.8
Radii of a polydisk      9.1
Radius of a ball      3.4
Rank theorem      10.3
Rational number      2.2
Real line      3.2
Real number      2.1
Real part of a complex number      4.4
Real vector space      5.1
Reflexivity of a relation      1.8
Regular frontier point for an analytic function      9.15 prob.
Regular value for an operator      11.1
Regularization      8.12 prob.
Regulated function      7.6
Relative maximum      3.9 prob.
Relatively compact set      3.17
Remainder (nth) of a series      5.2
Reproducing kernel      6.3 prob.
Residue      9.15
Resolvent of a linear differential equation      10.8
Restriction of a mapping      1.4
Riemann Sums      8.7 prob.
Riesz (F.)’s theorem      5.9
Road      9.6
Rolle’s Theorem      8.2 prob.
Rouche’s Theorem      9.17
scalar      9.1
Scalar product      6.2
Schoenflies’s theorem      9.Ap. prob.
Schottky’s theorem      10.3 prob.
Schwarz’s lemma      9.5 prob.
Second mean value theorem      8.7 prob.
Segment      5.1 prob. and
Self-adjoint operator      11.5
Semi-open interval      2.1
Separable metric space      3.10
Separating points (set of functions)      7.3
Separating two points (subset of the plane)      9.Ap.3
SEQUENCE      1.8
Series      5.2
Set      1.1
Set of mappings      1.4
Set of uniqueness for analytic functions      9.4
Simple arc, simple closed curve, simple loop, simple path      9.Ap.4
Simply connected domain      9.7 and 10.2 prob.6
Simply convergent sequence, simply convergent series      7.1
Simpson’s formula      8.14 prob.
Singular frontier point for an analytic function      9.15 prob.
Singular part of an analytic function at a point      9.15
Singular values of a compact operator      11.5 prob.15
Solution of a differential equation      10.4 and 11.7
Spectral value, spectrum of an operator      11.1
Sphere      3.4
Square root of a positive hermitian compact operator      11.5 prob.
Star-shaped domain      9.7
Step function      7.6
Stone — Weierstrass theorem      7.3
Strict relative maximum      3.9 prob.
Strictly convex function      8.5 prob.
Strictly decreasing, strictly increasing, strictly monotone      4.2
Strictly negative, strictly positive number      2.2
Sturm — Liouville problem      11.7
Subfamily      1.8
Subsequence      3.13
Subset      1.4
subspace      3.10
Subspace of a normed space      5.4
Substitution of power series in power series      9.2
Sum of a family of sets      1.8
Sum of a series      5.2
Sum of an absolutely summable family      5.3
Supremum of a set, of a function      2.3
Surjection, surjective mapping      1.6
Symmetric bilinear form      6.1
Symmetry of a relation      1.8
System of scalar linear differential equations      10.6
Tangent mappings at a point      8.1
Tauber’s theorem      9.3 prob.
Taylor’s Formula      8.14
Term (wth) of a series      5.2
Theorem of residues      9.16
Tietze-Urysohn extension theorem      4.5
Titchmarsh’s theorem      11.6 prob.
Topological direct sum, topological direct summand, topological supplement      5.4
Topological notion      3.12
Topologically equivalent distances      3.12
topology      3.12
Total derivative      8.1
Total subset      5.4
Totally disconnected set      3.19
Transcendental entire function      9.15 prob.
Transitivity of a relation      1.8
Transported distance      3.3
Triangle inequality      3.1 and 5.1
Trigonometric polynomials      7.4
Trigonometric system      6.5
Ultrametric inequality      3.8 prob.
Underlying real vector space      5.1
Uniformly continuous function      3.11
Uniformly convergent sequence, uniformly convergent series      7.1
Uniformly equicontinuous set      7.5 prob.
Uniformly equivalent distances      3.14
Union of a family of sets      1.8
union of two sets      1.2
Unit circle      9.5
Unit circle taken n times      9.8
Value of a mapping      1.4
Vector basis      5.9 prob.
Vector space      5.1
Volterra kernel      11.6 prob.
Weierstrass’s approximation theorem      7.4
Weierstrass’s decomposition      10.2 prob.
Weierstrass’s preparation theorem      9.17 prob.
Weierstrass’s theorem on essential singularities      9.15 prob.
Zero of an analytic function      9.15
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