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1.

図書

図書
坂井英太郎著
出版情報: 東京 : 共立出版, 1952  338p ; 21cm
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2.

図書

東工大
目次DB

図書
東工大
目次DB
小松勇作著
出版情報: 東京 : 共立出版, 1956.10  2, 221, 3p ; 19cm
シリーズ名: 共立全書 ; 117
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目次情報: 続きを見る
第1章 集合
   1.一般概念 1
   2.集合に関する演算 7
   3.極限集合 13
   4.空間 16
   5.点集合に関する諸定義 24
   6.Enの網系 30
   7.Borel集合 35
   8 被覆定理 37
第2章 測度
   9.測度問題 40
   10.外測度 41
   11.内測度と可測性 47
   12.可測集合 51
   13.極限集合の測度 56
   14.運動の保測性 62
   15.非可測集合 65
第3章 可測函数
   16. 連続性と半連続性 68
   17.Baire函数 74
   18.函数の可測性 77
   19.Borel 可測函数 84
   20 可測写像 85
第4章 Lebesgue積分
   21.縦線集合 89
   22.積分の定義 92
   23.函数の可積性と可測性 96
   24.和の極限としての積分 101
   25.積分の他の諸定義 108
   26.積分の性質 112
   27.極限函数の積分 117
   28.Fubiniの定理 126
   29.不定積分 132
   30.区間函数の拡大 136
   31.Riemann積分との比較 142
第5章 不定積分の微分
   32.許容集合列 151
   33.微分可能性 154
   34.Vitaliの被覆定理 157
   35.Lebesgueの定理 160
第6章 一変数の函数
   36.点函数と区間函数 165
   37.単調点函数 169
   38.値の和が有界な区間函数 174
   39.区間函数の微分 181
   40.導函数の性質 190
   41.原始函数と不定積分 197
   42.部分積分と変数の置換 200
   43.函数族Lp 207
   44.平均収斂 208
   45.Lebesgue-Stieltjes積分 211
   索引 222
第1章 集合
   1.一般概念 1
   2.集合に関する演算 7
3.

図書

図書
河野伊三郎著
出版情報: 東京 : 岩波書店, 1955.10  vii, 279p ; 18cm
シリーズ名: 岩波全書 ; 210
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4.

図書

図書
秋山武太郎著 ; 春日屋伸昌改訂
出版情報: 東京 : 日新出版, 1959.3  194, 2p ; 22cm
シリーズ名: わかる数学全書 ; 6
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5.

図書

図書
渡邊不二雄著
出版情報: 東京 : 宝文館, 1950  冊 ; 22cm
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6.

図書

図書
三村征雄編
出版情報: 東京 : 裳華房, 1955.11  viii, 388p ; 22cm
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7.

図書

図書
福原満洲雄, 山崎三郎編著
出版情報: 東京 : 共立出版, 1954.4  237p ; 21cm
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8.

図書

図書
奥川光太郎, 桑垣煥, 河合良一郎共編
出版情報: 東京 : 朝倉書店, 1956.7  4, 430p ; 22cm
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9.

図書

図書
平野幸太郎著
出版情報: 東京 : 森北出版, 1957.12  iii, 292p ; 21cm
シリーズ名: 理論・応用数学演習全書 / 矢野健太郎責任編集 ; 2
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10.

図書

図書
by Hassler Whitney
出版情報: Princeton, N.J. : Princeton University Press, 1957  xv, 387 p ; 24 cm
シリーズ名: Princeton mathematical series ; 21
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目次情報: 続きを見る
Preface
Introduction
The general problem of integration / A:
The integral as a function of the domain / 1:
Polyhedral chains / 2:
Two continuity hypotheses / 3:
A further continuity hypothesis / 4:
Some examples / 5:
The case r = n / 6:
The r-vector of an oriented r-cell / 7:
On r-vectors and boundaries of (r + 1)-cells / 8:
Grassmann algebra / 9:
The dual algebra / 10:
Integration of differential forms / 11:
Some classical topics / B:
Grassmann algebra in metric oriented n-space / 12:
The same, n = 3 / 13:
The differential of a mapping / 14:
Jacobians / 15:
Transformation of the integral / 16:
Smooth manifolds / 17:
Particular forms of integrals in 3-space / 18:
The Theorem of Stokes / 19:
The exterior differential / 20:
Some special formulas in metric oriented E[superscript 3] / 21:
An existence theorem / 22:
De Rham's Theorem / 23:
Indications of general theory / C:
Normed spaces of chains and cochains / 24:
Continuous chains / 25:
On 0-dimensional integration / 26:
Classical theory / Part I:
Multivectors / Chapter I:
Multicovectors
Properties of V[subscript r] and V[superscript r]
Alternating r-linear functions
Use of coordinate systems
Exterior products
Interior products
n-vectors in n-space
Simple multivectors
Linear mappings of vector spaces
Duality
Euclidean vector spaces
Mass and comass
Mass and comass of products
On projections
Differential forms / Chapter II:
The differential of a smooth mapping
Some properties of differentials
Smooth mappings
The inverse and implicit function theorems
A representation of vectors and covectors
The tangent space of a smooth manifold
Differential forms in smooth manifolds
A characterization of the exterior differential
Riemann integration theory / Chapter III:
The r-vector of an oriented r-simplex
The r-vector of an r-chain
Integration over cellular chains
Some properties of integrals
Relation to the Riemann integral
Integration over open sets
The transformation formula
Proof of the transformation formula
Transformation of the Riemann integral
Integration in manifolds
Stokes' Theorem for a parallelepiped
A special case of Stokes' Theorem
Sets of zero s-extent
Stokes' Theorem for standard domains
Proof of the theorem
Regular forms in Euclidean space
Regular forms in smooth manifolds
Stokes' Theorem for standard manifolds
The iterated integral in Euclidean space
Manifolds in Euclidean space / Chapter IV:
The imbedding theorem
The compact case
Separation of subsets of E[superscript m]
Regular approximations
Proof of Theorem 1A, M compact
Admissible coordinate systems in M
Proof of Theorem 1A, M not compact
Local properties of M in E[superscript m]
On n-directions in E[superscript m]
The neighborhood of M in E[superscript m]
Projection along a plane
Triangulation of manifolds
The triangulation theorem
Outline of the proof
Fullness
Linear combinations of edge vectors of simplexes
The quantities used in the proof
The complex L
The complex L*
The intersections of M with L*
The complex K
Imbedding of simplexes in M
The complexes K[subscript p]
Cohomology in manifolds
[mu]-regular forms
Closed forms in star shaped sets
Extensions of forms
Elementary forms / 27:
Certain closed forms are derived / 28:
Isomorphism of cohomology rings / 29:
Periods of forms / 30:
The Hopf invariant / 31:
On smooth mappings of manifolds / 32:
Other expressions for the Hopf invariant / 33:
General theory / Part II:
Abstract integration theory / Chapter V:
Mass of polyhedral chains
The flat norm
Flat cochains
Examples
The sharp norm
Sharp cochains
Characterization of the norms
An algebraic criterion for a multicovector
Sharp r-forms
The semi-norms [vertical bar A vertical bar superscript flat], [vertical bar A vertical bar superscript sharp] are norms
Weak convergence
Some relations between the spaces of chains and cochains
The [rho]-norms
The mass of chains
Separability of spaces of chains
Non-separability of spaces of cochains
Some relations between chains and functions / Chapter VI:
Continuous chains on the real line
0-chains in E[superscript 1] defined by functions of bounded variation
Sharp functions times 0-chains
The part
Functions of bounded variation in E[superscript 1] defined by 0-chains
Some related analytical theorems
Continuous r-chains in E[superscript n]
On compact cochains
The boundary of a smooth chain
Continuous chains in smooth manifolds
General properties of chains and cochains / Chapter VII:
Sharp functions times chains
Sharp functions times cochains
Supports of chains and cochains
On non-compact chains
On polyhedral approximations
Sharp chains at a point
Molecular chains are dense
Flat r-chains in E[superscript r-k] are zero
Flat cochains in complexes
Elementary flat cochains in a complex
The isomorphism theorem
Chains and cochains in open sets / Chapter VIII:
Chains and cochains in open sets, elementary properties
Chains and cochains in open sets, further properties
Properties of mass
On the open sets to which a chain belongs
An expression for flat chains
An expression for sharp chains
Lebesgue theory / Part III:
Flat cochains and differential forms / Chapter IX:
n-cochains in E[superscript n]
Some properties of fullness
Properties of projections
Elementary properties of D[subscript X](p, [alpha])
The r-form defined by a flat r-cochain
Flat r-forms
Flat r-forms and flat r-cochains
Flat r-direction functions
Flat forms defined by components
Approximation to D[subscript X](p) by [characters not reproducible]
Total differentiability of Lipschitz functions
On the exterior differential of r-forms
On averages of r-forms
Products of cochains
Lebesgue chains
Products of cochains and chains
Products and weak limits
Characterization of the procducts
Lipschitz mappings / Chapter X:
Affine approximations to Lipschitz mappings
The approximation on the edges of a simplex
Approximation to the Jacobian
The volume of affine approximations
A continuity lemma
Lipschitz chains
Lipschitz mappings of open sets
Lipschitz mappings and flat cochains
Lipschitz mappings and flat forms
Lipschitz mappings and sharp functions
Lipschitz mappings and products
On the flat norm of Lipschitz chains
Deformations of chains
Chains and additive set functions / Chapter XI:
On finite dimensional Banach spaces
Vector valued additive set functions
Vector valued integration
Point functions times set functions
Relations between a set function and its variation
On positive linear functionals
On bounded linear functionals
Linear functions of sharp r-forms
The sharp norm of r-vector valued set functions
Molecular set functions
Sharp chains and set functions
Bounded Borel functions times chains
The part of a chain in a Borel set
Chains and point functions
Characterization of the sharp norm
Expression for the sharp norm
Other expressions for the norm
Vector and linear spaces / Appendix I:
Vector spaces
Linear transformations
Conjugate spaces
Direct sums, complements
Quotient spaces
Pairing of linear spaces
Abstract homology
Normed linear spaces
Euclidean linear spaces
Affine spaces
Barycentric coordinates
Affine mappings
Euclidean spaces
Banach spaces
Semi-conjugate spaces
Geometric and topological preliminaries / Appendix II:
Cells, simplexes
Polyhedra, complexes
Su bdivisions
Standard subdivisions
Orientation
Chains and cochains
Boundary and coboundary
Homology and cohomology
Products in a complex
Joins
Subdivisions of chains
Cartesian products of cells
Mappings of complexes
Some properties of planes
Mappings of n-pseudomanifolds into n-space
Distortion of triangulations of E[superscript m]
Analytical preliminaries / Appendix III:
Existence of certain functions
Partitions of unity
Smoothing functions by taking averages
The Weierstrass approximation theorem
The space L[superscript 1]
Index of symbols
Index of terms
Preface
Introduction
The general problem of integration / A:
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