Analysis: Part II Integration, Distributions, Holomorphic Functions, Tensor and Harmonic Analysis

Description

The extraordinarily rapid advances made in mathematics since World War II have resulted in analysis becoming an enormous organism spread­ ing in all directions. Gone for good surely are the days of the great French "courses of analysis" which embodied the whole of the "ana­ lytical" knowledge of the times in three volumes-as the classical work of Camille Jordan. Perhaps that is why present-day textbooks of anal­ ysis are disproportionately modest relative to the present state of the art. More: they have "retreated" to the state before Jordan and Goursat. In recent years the scene has been changing rapidly: Jean Dieudon­ ne is offering us his monumentel Elements d'Analyse (10 volumes) written in the spirit of the great French Course d'Analyse. To the best of my knowledge, the present book is the only one of its size: starting from scratch-from rational numbers, to be precise-it goes on to the theory of distributions, direct integrals, analysis on com­ plex manifolds, Kahler manifolds, the theory of sheaves and vector bun­ dles, etc. My objective has been to show the young reader the beauty and wealth of the unsual world of modern mathematical analysis and to show that it has its roots in the great mathematics of the 19th century and mathematical physics. I do know that the young mind eagerly drinks in beautiful and difficult things, rejoicing in the fact that the world is great and teeming with adventure. Front Matter....Pages i-xvii Topology Uniform Structures Function Spaces....Pages 1-61 Theory of the Integral....Pages 63-198 Tensor Analysis Harmonic Forms Cohomology Applications to Electrodynamics....Pages 199-332 Elementary Properties of Holomorphic Functions of Several Variables Harmonic Functions....Pages 333-384 Complex Analysis in One Dimension (Riemann Surfaces)....Pages 385-649 Normal and Paracompact Spaces Partition of Unity....Pages 651-686 Measurable Mappings the Transport of a Measure Convolutions of Measures and Functions....Pages 687-705 The Theory of Distributions Harmonic Analysis....Pages 707-794 Back Matter....Pages 795-829