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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.


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

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Предметный указатель
$\varepsilon$-Capacity of a set      3.16 prob.
$\varepsilon$-Entropy of a set      3.16 prob.
Abel’s lemma      9.1
Abel’s theorem      9.3 prob.
Absolute value of a complex number      4.4
Absolute value of a real number      2.2
Absolutely convergent series      5.3
Absolutely summable family, absolutely summable subset      5.3
Adjoint of an operator      11.5
Algebraic multiplicity of an eigenvalue      11.4
Amplitude of a complex number      9.5 prob.
Analytic mapping      9.3
Approximate solution of a differential equation      10.5
Ascoli’s Theorem      7.5
At most denumerable set, at most denumerable family      1.9
Axiom of Archimedes      2.1
Axiom of Choice      1.4
Axiom of nested intervals      2.1
Banach space      5.1
Basis for the open sets of a metric space      3.9
Belonging to a set      1.1
Bergman’s kernel      9.13 prob.
Bessel’s inequality      6.5
Bicontinuous mapping      3.12
Bijective mapping, bijection      1.6
Bloch’s constant      10.3 prob.
Bolzano’s theorem      3.19
Borel — Lebesgue axiom      3.16
Borel — Lebesgue theorem      3.17
Borel’s theorem      8.14 prob.
Boundary conditions for a differential equation      11.7
Bounded from above, from below (subset of R)      2.3
Bounded real function      2.3
Bounded set in a metric space      3.4
Bounded subset of R      2.3
Broken line      5.1 prob.
Brouwer’s theorem for the plane      10.2 prob.
Canonical decomposition of a vector relatively to a hermitian compact operator      11.5
Cantor’s triadic set      4.2 prob.
Cartesian product of sets      1.3
Cauchy criterion for sequences      3.14
Cauchy criterion for series      5.2
Cauchy sequence      3.14
Cauchy — Schwarz inequality      6.2
Cauchy’s conditions for analytic functions      9.10
Cauchy’s existence theorem for differential equations      10.4
Cauchy’s formula      9.9
Cauchy’s inequalities      9.9
Cauchy’s theorem on analytic functions      9.6
Center of a ball      3.4
Center of a polydisk      9.1
Change of variables in an integral      8.7
Circuit      9.6
Closed ball      3.4
Closed interval      2.1
Closed polydisk      9.1
Closed set      3.8
Closure of a set      3.8
Cluster point of a set      3.8
Cluster value of a sequence      3.13
Codimension of a linear variety      5.1 prob.
Coefficient (nth) with respect to an orthonormal system      6.5
Commutatively convergent series      5.3 prob.
Compact operator      11.2
Compact set      3.17
Compact space      3.16
Complement of a set      1.2
Complete space      3.14
Complex number      4.4
Complex vector space      5.1
Composed mapping      1.7
Condensation point      3.9 prob.
Conformal mapping theorem      10.3 prob.
Conjugate of a complex number      4.4
Connected component of a set, of a point in a space      3.19
Connected set, connected space      3.19
Constant mapping      1.4
Contained in a set, containing a set      1.1
Continuity of the roots as function of parameters      9.17
Continuous, continuous at a point      3.11
Continuously differentiable mapping      8.9
Convergence radius of a power series      9.1 prob.1
Convergent sequence      3.13
Convergent Series      5.2
Convex set, convex function      8.5 prob.
Coordinate (nth) with respect to an orthonormal system      6.5
Covering of a set      1.8
Cross section of a set      1.3
Cut of the plane      9.Ap.3
Decreasing function      4.2
Degenerate hermitian form      6.1
Dense set in a space, dense set with respect to another set      3.9
Denumerable set, denumerable family      1.9
Derivative (pth) with respect to an interval      8.12
Derivative (second, pth)      8.12
Derivative in an open set      8.1
Derivative of a function of one variable      8.4
Derivative of a mapping at a point      8.1
Derivative on the left, on the right      8.4
Derivative with respect to a subset of $\mathbf{R}$      8.4
Diagonal      1.4
Diagonal process      9.13
Diameter of a set      3.4
Difference of two sets      1.2
Differentiable (twice, p times)      8.12
Differentiable mapping at a point, in a set      8.1
Differentiable with respect to the first, second,..., variable      8.9
Differential equation      10.4
Dimension of a linear variety      5.1 prob.
Dini’s theorem      7.2
Direct image      1.5
Dirichlet’s function      3.11
Discrete metric space      3.2 and 3.12
Disk      4.4
Distance of two points      3.1
Distance of two sets      3.4
Eigenfunction of a kernel function      11.6
Eigenspace corresponding to an eigenvalue      11.1
Eigenvalue of a Sturm — Liouville problem      11.7
Eigenvalue of an operator      11.1
Eigenvector of an operator      11.1
Eilenberg’s criterion      9.Ap.3
Element      1.1
Elementary solution for a Sturm — Liouville problem      11.7
Empty set      1.1
Endless road      9.12 prob.
Entire function      9.3
Equation of a hyperplane      5.8
Equicontinuous at a point, equicontinuous      7.5
Equipotentsets      1.9
Equivalence class, equivalence relation      1.8
Equivalent norms      5.6
Equivalent roads      9.6
Essential mapping      9.Ap.2
Essential singular point, essential singularity      9.15
Euclidean distance      3.2
Everywhere dense set      3.9
Exponential function      4.3 and 9.5
Extended real line      3.3
Extension of a mapping      1.4
Exterior point of a set, exterior of a set      3.7
Extremity of a path      9.6
Extremity of an interval      2.1
Family of elements      1.8
Finer distance, finer topology      3.12
Finite number      3.3
Fixed point theorem      10.1
Fourier coefficient (nth)      6.5
Fredholm equation, Fredholm alternative      11.6
Frobenius — Perron’s theorem      11.1 prob.
Frobenius’s Theorem      10.9
Frontier point of a set, frontier of a set      3.8
Full sequence of positive eigenvalues      11.5 prob.
Function      1.4
Function of bounded variation      7.6 prob.
Function of positive type      6.3 prob.
Functional graph, functional relation      1.4
Functions coinciding in a subset      1.4
Fundamental system of neighborhoods      3.6
Fundamental theorem of algebra      9.11
Geometric multiplicity of an eigenvalue      11.4
Goursat’s theorem      9.10 prob.
Gram determinant      6.6 prob.
Graph of a mapping      1.4
Graph of a relation      1.3
Greatest lower bound      2.3
Green function of a Sturm — Liouville problem      11.7
Gronwall’s lemma      10.5
Haar orthonormal system      8.7 prob.
Hadamard’s gap theorem      9.15 prob.
Hadamard’s three circles theorem      9.5 prob.
Hausdorff distance of two sets      3.16 prob.
Hermitian form      6.1
Hermitian kernel      11.6
Hermitian norm      9.5 prob.
Hermitian operator      11.5
Hilbert basis      6.5
Hilbert space      6.2
Hilbert sum of Hilbert spaces      6.4
Homeomorphic metric spaces, homeomorphism      3.12
Homo topic paths, homotopic loops, homotopy of a path into a path      9.6 and 10.2 prob.
Homogeneous hyperplane      5.8 prob.
Homogeneous linear differential equation      10.8
Hyperplane      5.8 and 5.8 prob.
Hyperplane of support      5.8 prob.
Identity mapping      1.4
Image of a set by a mapping      1.5
Imaginary part of a complex number      4.4
Implicit function theorem      10.2
Improperly integrable function along an endless road, improper integral      9.12 prob.
Increasing function      4.2
Increasing on the right      8.5 prob.
Indefinitely differentiable mapping      8.12
Index of a point with respect to a circuit, of a circuit with respect to a point      9.8
Index of a point with respect to a loop      9.Ap.1
Induced distance      3.10
Inessential mapping      9.Ap.2
Infimum of a set, of a function      2.3
Infinite product of metric spaces      3.20 prob.
Injection, injective mapping      1.6
Integer (positive or negative)      2.2
integral      8.7
Integral along a road      9.6
Integration by parts      8.7
Interior point of a set, interior of a set      3.7
Intersection of a family of sets      1.8
intersection of two sets      1.2
Inverse image      1.5
Inverse mapping      1.6
Isolated point of a set      3.10
Isolated singular point      9.15
Isometric spaces, isometry      3.3
Isomorphism of prehilbert spaces      6.2
Isotropic vector      6.1
Jacobian matrix, jacobian      8.10
Janiszewski’s theorem      9.Ap.3
Jordan curve theorem      9.Ap.4
Juxtaposition of two paths      9.6
Kernel function      11.6
Lagrange’s inversion formula      10.2 prob.
Laurent series      9.14
Least upper bound      2.3
Lebesgue function (nth)      11.6 prob.
Lebesgue’s property      3.16
Legendre polynomials      6.6 and 8.14 prob.
Leibniz’s formula      8.13
Leibniz’s Rule      8.11
Length of an interval      2.2
Limit of a function, limit of a sequence      3.13
Limit on the left, limit on the right      7.6
Linear differential equation      10.6
Linear differential equation of order n      10.6
Linear differential operator      8.13
Linear form      5.8
Linear variety      5.1 prob.
Linked by a broken line (points)      5.1 prob.
Liouville’s theorem      9.11
Lipschitzian function      7.5 prob. and
Locally closed set      3.10 prob.
Locally compact space      3.18
Locally connected space      3.19
Locally Lipschitzian function      10.4
Logarithm      4.3 and 9.5 prob.
Loop      9.6 and 10.2 prob.
Loop homotopy      9.6 and 10.2 prob.
Majorant      2.3
Majorized set, majorized function      2.3
Mapping      1.4
Maximal solution of a differential equation      10.7 prob.
Maximinimal principle      11.5 prob. and prob.
Mean value theorem      8.5
Mercer’s theorem      11.6
Meromorphic function      9.17
Method of the gliding hump      11.5 prob. and prob.
Metric space      3.1
Minimal solution of a differential equation      10.7 prob.
Minkowski’s inequality      6.2
Minorant      2.3
Minorized set, minorized function      2.3
Monotone function      4.2
Morera’s theorem      9.10 prob.
Natural boundary      9.15 prob.
Natural injection      1.6
Natural mapping of X into X/R      1.8
Natural ordering      2.2
Negative number      2.2
Negative real half-line      9.5 prob.
Neighborhood      3.6
Newton’s approximation method      10.2 prob.
Nondegenerate hermitian operator      11.5
Norm      5.1
Normally convergent series, normally summable family      7.1
Normed space      5.1
One-to-one mapping      1.6
Onto mapping      1.6
Open ball      3.4
Open covering      3.16
Open interval      2.1
Open neighborhood      3.6
Open polydisk      9.1
Open set      3.5
Operator      11.1
Opposite path      9.6
Order of a linear differential operator      8.13
Order of an analytic function at a point      9.15
Ordered pair      1.3
Origin of a path      9.6
Origin of an interval      2.1
Orthogonal projection      6.3
Orthogonal supplement      6.3
Orthogonal system      6.5
Orthogonal to a set (vector)      6.1
Orthogonal vectors      6.1
Orthonormal system      6.5
Orthonormalization      6.6
Oscillation of a function      3.14
p-adic distance      3.2
Parallel hyperplane      5.8 prob.
Parseval’s identities      6.5
Partial derivative      8.9
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