Le caractere propre des methodes de l'Analyse et de
la Geometrie modernes consiste dans l'emploi d'un
petit nombre de principes generaux, independants
de la situation respective des differentes parties ou
des valeurs relatives des differents symboles; et les
consequences sont d'autant plus etendues que les
principes eux-memes ont plus de generalite.
from G.
Principes de Geometrie Analy-
This text is written for the graduate student who has previous training
in analysis and linear algebra, as for instance S. Lang's Analysis / a n d Lin-
ear Algebra. It is meant as an introduction to what is today an intensive
area of research linking several disciplines of mathematics and physics in
the sense of the Greek word avinrXeKStu (which means to interconnect, or
to interrelate in
The difficulty (but also the fascination) of the
area is the wide variety of mathematical machinery required. In order to
introduce this interrelation, this text includes extensive appendices which
include definitions and developments not usually covered in the basic train-
ing of students but which lay the groundwork for the specific constructions
I want to thank P. Slodowy for pointing out to me that the name symplectic group, which
eventually gave rise to the term symplectic geometry, was proposed by H. WEYL, [W], 1938, in
his book, The Classical Groups (see footnote on p. 165). The symplectic group was also called
the complex group or an Abelian linear group, this last to honor ABEL, who was the first to study
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