دانلود کتاب Integration of Functionals. Seminar

The present notes contain a report of lectures given in a seminar on Functional Analysis, held in the Spring of 1955 at the Institute of Mathematical Sciences. The general topic of this seminar was יי integration in functional space״, with the main emphasis placed upon the integration of functionals defined over a Hilbert space. The first chapters (I, II, VI, VIII) of the lectures consist of an elaboration of an approach to this problem indicated in an earlier work of one of the contributors. In pursuing this, a result on the invariance of integrals of cylinder functionals was obtained, independently and at the same time, by J. Schwartz and by K.O. Friedrichs and H.N. Shapiro (Chapters VI, VII). This result in turn led to an extension of the Gaussian measure over Hilbert space into an ״outer Hilbert Space״ in which the extended measure is totally additive. (Chapter XII*) The notes include some discussion (Chapters IX, X) of the structural relationship of the integral considered here to the Wiener integral. It should also be noted that Bochner’s extension of the Kolmogoroff Theorem ** is similar in spirit to the ideas developed here. However, Bochner's Theorem does not cover the situation considered in the notes. In the present notes no discussion is included of the work of I.E. Segal in this area. The seminar was directed by K.O. Friedrichs and H.N. Shapiro and assisted by Thomas Seidman and Charles Lytle.



A preliminary report on these results has appeared in the

*Proceedings of the National Academy of Sciences, Vol. 43, No. 4,
pp. 336-338, April 1957.

** S. Bochner, Harmonic Analysis and the Theory of Probability ,
Chapter v. University of California Press, 1955•

*** This work was supported by the Office of Naval Research and
the National Science Foundation.