It seems to be well forgotten that many of the first ideas in geometry, basis theory
and isomorphic theory of Banach spaces have vector measure-theoretic origins.
Equally well forgotten is the fact that much of the early interest in weak and weak*
compactness was motivated by vector measure-theoretic considerations.
In 1936, J. A. Clarkson introduced the notion of uniform convexity to prove that
absolutely continuous functions on a Euclidean space with values in a uniformly
convex Banach space are the integrals of their derivatives. At the same time,
Clarkson used vector measure-theoretic ideas to prove that many familiar Banach
spaces do not admit equivalent uniformly convex norms.
N. Dunford and A. P. Morse, in 1936, introduced the notion of a boundedly
complete basis to prove that absolutely continuous functions on a Euclidean space
with values in a Banach space with a boundedly complete basis are the integrals of
their derivatives. Shortly thereafter Dunford was able to recognize the Dunford-
Morse theorem and the Clarkson theorem as genuine Radon-Nikodym theorems
for the Bochner integral. This was the first Radon-Nikodym theorem for vector
measures on abstract measure spaces.
B. J. Pettis, in 1938, made his contribution to the Orlicz-Pettis theorem for the
purpose of proving that weakly countably additive vector measures are norm
In 1938,1. Gel'fand used vector measure-theoretic methods to prove that L^O, 1]
is not isomorphic to a dual of a Banach space.
In 1939, Pettis showed that the notions of weak and weak* compactness are
intimately related to the problem of differentiating vector-valued functions on
Euclidean space. Dunford and Pettis, in 1940, built on their earlier work to repre-
sent weakly compact operators on L\ and the general operator from Lx to a
separable dual space by means of a Bochner integral. By means of their integral
representation they were able to prove that Lx has the property now known as the
Then came the war. By the end of the war, the love affair between vector measure
theory and Banach space theory had cooled. They began to drift down separate
paths. Neither prospered. Much of Banach space theory became lost in the mazes
of the theory of locally convex spaces. The work in vector measure theory became