Vol. 59, No. 2, 1975

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On conjugate Banach spaces with the Radon-Nikodým property

Tsang Hai Kuo

Vol. 59 (1975), No. 2, 497–503

It is shown that if the unit ball BX∗∗ of X∗∗ is Eberlein compact in the weaktopology, or if xis isomorphic to a subspace of a weakly compactly generated Banach space then xpossesses the Radon-Nikodým property (RNP). This extends the classical theorem of N. Dunford and B. J. Pettis. If X is a Banach space with X∗∗∕X separable then both xand x ∗∗ (and hence X) have the RNP. It is also shown that if a conjugate space X possesses the RNP and X is weaksequentially dense in X∗∗ then BX∗∗ is weaksequentially compact. Thus, in particular, if x ∗∗∕X is separable then BX∗∗∗ is weaksequentially compact.

Mathematical Subject Classification 2000
Primary: 46B10
Secondary: 28A45
Received: 24 April 1975
Published: 1 August 1975
Tsang Hai Kuo