Many areas of applied mathematics call for an efficient calculus in
infinite dimensions. This is most apparent in quantum physics and in all
disciplines of science which describe natural phenomena by equations
involving stochasticity. With this monograph we intend to provide a
framework for analysis in infinite dimensions which is flexible enough
to be applicable in many areas, and which on the other hand is intuitive
and efficient. Whether or not we achieved our aim must be left to the
judgment of the reader. This book treats the theory and applications of
analysis and functional analysis in infinite dimensions based on white
noise. By white noise we mean the generalized Gaussian process which is
(informally) given by the time derivative of the Wiener process, i.e.,
by the velocity of Brownian mdtion. Therefore, in essence we present
analysis on a Gaussian space, and applications to various areas of
sClence. Calculus, analysis, and functional analysis in infinite
dimensions (or dimension-free formulations of these parts of classical
mathematics) have a long history. Early examples can be found in the
works of Dirichlet, Euler, Hamilton, Lagrange, and Riemann on
variational problems. At the beginning of this century, Frechet, Gateaux
and Volterra made essential contributions to the calculus of functions
over infinite dimensional spaces. The important and inspiring work of
Wiener and Levy followed during the first half of this century.
Moreover, the articles and books of Wiener and Levy had a view towards
probability theory.
