Mathematical anaysis during the 20th century

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Mathematical anaysis during the 20th century / Jean-Paul Pier

資料種別:: 図書
出版情報:: Oxford : Oxford University Press, 2001
形態:: x, 428 p. ; 25 cm
著者名:: Pier, Jean-Paul, 1933- <DA01789715>
ISBN:: 9780198503941 [0198503946] (: hbk)
書誌ID:: BA53431252

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注記:

Includes bibliographical references (p. [350]-417) and index

主題:

Mathematics--History--20th century

言語:

英語

目次情報:

Introduction / 1：

The scope of analysis / 1.1：

The great classics on analysis / 1.1.1：

The changing object of analysis / 1.1.2：

Main streams in a turbulent activity / 1.2：

The question of subdividing mathematical analysis / 1.2.1：

How to organize the subject / 1.2.2：

General Topology / 2：

Evolution 1900-1950 / 2.1：

Topological axiomatizations / 2.1.1：

Topological algebra / 2.1.2：

Filtrations / 2.1.3：

Dimension theory / 2.1.4：

Complementary inputs / 2.1.5：

Flashes 1950-2000 / 2.2：

An accomplished subject / 2.2.1：

Generalized topological concepts / 2.2.2：

Integration and Measure / 3：

Lebesgue integration / 3.1：

The general concept of measure / 3.1.2：

Paradoxical decomposition / 3.1.3：

Period of consolidation / 3.1.4：

Standing problems / 3.2：

Abstract formulations / 3.2.2：

Generalized Riemann integrals / 3.2.3：

Outlook / 3.2.4：

Functional analysis / 4：

New objectives / 4.1：

Theory of integral equations / 4.1.2：

Banach spaces / 4.1.3：

Hilbert spaces / 4.1.4：

von Neumann algebras / 4.1.5：

Banach algebras / 4.1.6：

Distributions / 4.1.7：

Topological vector spaces / 4.2：

Extension of Weierstra[beta]'s theorem / 4.2.2：

Frechet spaces, Schwartz spaces, Sobolev spaces / 4.2.3：

Banach space properties / 4.2.4：

Hilbert space properties / 4.2.5：

Banach algebra and C*-algebra properties / 4.2.6：

Approximation properties / 4.2.7：

Nuclearity / 4.2.8：

von Neumann algebra properties / 4.2.9：

Specific topics / 4.2.10：

Harmonic analysis / 5：

Fourier series / 5.1：

Invariant measures / 5.1.2：

Almost periodic functions / 5.1.3：

Uniqueness of invariant measures / 5.1.4：

Convolutions / 5.1.5：

An evolution linked to the history of physics / 5.1.6：

Representation theory / 5.1.7：

Structural properties of topological groups / 5.1.8：

Positive-definite functions / 5.1.9：

Harmonic synthesis / 5.1.10：

Metric locally compact Abelian groups / 5.1.11：

Fourier transforms / 5.2：

Convolution properties / 5.2.2：

Group representations / 5.2.3：

Remarkable Banach algebras of functions on a locally compact group / 5.2.4：

Specific sets / 5.2.5：

Specific groups / 5.2.6：

Harmonic analysis on semigroups / 5.2.7：

Wavelets / 5.2.8：

Generalized actions / 5.2.9：

Lie groups / 6：

Lie groups and Lie algebras / 6.1：

Symmetric Riemannian spaces / 6.1.2：

Hilbert's problem for Lie groups / 6.1.3：

Representations of Lie groups / 6.1.4：

The wide range of Lie group theory / 6.2：

Solution of Hilbert's problem on Lie groups / 6.2.2：

Ergodicity problems / 6.2.3：

Specific classes of Lie groups / 6.2.4：

Extensions of Lie group theory / 6.2.5：

Theory of functions and analytic geometry / 7：

The nineteenth century continued / 7.1：

Potential theory / 7.1.2：

Conformal mappings / 7.1.3：

Towards a theory of several complex variables / 7.1.4：

Accomplishments on previous topics / 7.2：

Hardy spaces / 7.2.2：

The dominance of the theory of several complex variables / 7.2.3：

Iteration problems / 7.2.4：

Ordinary and Partial Differential Equations / 8：

New trends for classical problems / 8.1：

Fixed point properties / 8.1.2：

From the ordinary differential case to the partial differential case / 8.1.3：

Differential equations / 8.2：

Partial differential equations / 8.2.2：

Tentacular subjects / 8.2.3：

Algebraic topology / 9：

The origins of algebraic topology / 9.1：

Simplicial theories / 9.1.2：

Homotopy theory / 9.1.3：

Fibres and fibrations / 9.1.4：

The breakthroughs due to Eilenberg, MacLane, and Leray / 9.1.5：

The power of the machinery / 9.2：

Generalizations / 9.2.2：

Differential topology / 10：

The beginning of the century / 10.1：

E. Cartan's work / 10.1.2：

Tensor products and exterior differentials / 10.1.3：

Morse theory / 10.1.4：

Whitney's work / 10.1.5：

De Rham's work / 10.1.6：

Hodge theory / 10.1.7：

The framing of the subject / 10.1.8：

The status of differentiable manifolds / 10.2：

Foliations / 10.2.2：

From Poincare's heritage / 10.2.3：

Global analysis / 10.2.5：

Probability / 11：

First results / 11.1：

Brownian motion / 11.1.2：

Ergodicity / 11.1.3：

Probabilities as measures / 11.1.4：

Stochastic integrals / 11.1.5：

Probability theory, a part of analysis / 11.2：

Dynamical systems and ergodicity / 11.2.2：

Entropy / 11.2.3：

Stochastic processes / 11.2.4：

Algebraic geometry / 12：

Algebraic geometry and number theory / 12.1：

The Mordell conjecture / 12.1.2：

Transcendence and prime numbers / 12.1.3：

The Riemann conjecture / 12.1.4：

Arithmetical properties / 12.2：

Investigations on transcendental numbers / 12.2.2：

A central object of study / 12.2.3：

Etale cohomology / 12.2.4：

The general Riemann-Roch theorems / 12.2.5：

K-theory / 12.2.6：

Further studies / 12.2.7：

References

Index of Names

Index of Terms

List of Symbols / Appendix：

Introduction / 1：

The scope of analysis / 1.1：

The great classics on analysis / 1.1.1：

The changing object of analysis / 1.1.2：

Main streams in a turbulent activity / 1.2：

The question of subdividing mathematical analysis / 1.2.1：

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