Much of the work on Banach spaces done in the 1930's resulted from investigat-
ing how much of real variable theory might be extended to functions taking values
in such spaces. Members of E. H. Moore's school of general analysis at Chicago,
including Graves and Hildebrandt, and functional analysts in Italy and Poland
(Orlicz in particular) had already done pioneer work in convergence of functions,
certain aspects of integration and differentiation, and the relationships between
various convergence properties for series. In the 1930's Hildebrandt's group in Ann
Arbor and Tamarkin's at Brown expanded the effort in the U.S.A., the strong
Russian school developed, and the influence of the Polish group spread, via
Banach's book, more deeply and widely. In developing integration and differentia-
tion theory for functions defined on Euclidean space to a Banach space B in the
period subsequent to Bochner's 1933 papers the important pioneer figures were
Dunford and Gel'fand.
It was in the study of differentiation of functions on Euclidean figures that the
role of the character of B emerged. Although some functions, such as Bochner
integrals, were differentiate a.e. regardless of B, many were not, their differen-
tiability depending on the characteristics of their range spaces; more precisely, it
depended on what properties the function developed for its range set as a subset of
B. (Clarkson invented uniformly convex spaces for the purpose of universal differ-
entiation; reflexive spaces reappeared on the stage for the same purpose.) More-
over differentiation, aside from its intrinsic interest, was fundamental in efforts
to represent linear operators by means of integrals, and when operations from
spaces of functions whose domains were an abstract space were to be represented,
differentiation had to be replaced by Radon-Nikodym theorems. Here Dunford led
by proving the earliest R-N theorem (N. Dunford, Integration and linear operations,
Trans. Amer. Math. Soc. 40 (1936), 474-494) and by giving the first proof of a
R-N theorem, now well known, when B is a dual space (N. Dunford and B. J.
Pettis, Linear operations on summable functions, Trans. Amer. Math. Soc. 47 (1940),
323-392; second proof). The study of Banach-space-valued functions waned in the
1940's, was revived and partly redirected by the deep work of Grothendieck, and
generally relapsed again until late in the 1960's. Since then vigorous work by many
here and in various parts of Europe and elsewhere has produced a flourishing body
Previous Page Next Page