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In this paper an example is given of a separable Banach space with the Radon-Nikodym property that is not isomorphic to a subspace of a separable dual space. In [5] Uhl showed that if every separable closed linear subspace of a Banach space X is isomorphic to a subspace of a separable dual, then X has the Radon-Nikodyim property (RNP). In that same paper, he asked whether every separable Banach space X with the RNP is isomorphic to a subspace of a separable dual space. Three years later, in 1975, Stegall [4] showed that if X is a subspace of a dual space, then the answer to Uhl's question is yes. We present in this note an example which shows that for general Banach spaces the answer is no. An example of a separable Banach space that has the RNP and is not isometric (but is isomorphic) to a subspace of a separable dual is given in [2] and [3]. Our construction employs this example as well as other results obtained in these papers. We will state the definitions and results to be used in this note and refer the reader to [2] for the details. Let X be a Banach space and let e be a positive number. Let S = f(k,j): k e N,j E N, 1 E. A finite tree with separation constant e is a function T:Sr, X that satisfies conditions (i), (ii) and (iii). It is known that if a Banach space X contains a tree, then X does not have the Radon-Nikodym property [1, p. 127]. The Banach space X is said to have the (e, 8)-neighborly tree property (NTP) if there are positive numbers e and 8 for which 8 < e/4 and the closed unit ball of X contains a sequence (Tn) of finite trees satisfying: (i) Each Tn is defined on Sn. (ii) Each Tn has separation constant e. Presented to the Society October 16, 1978; received by the editors January 25, 1979. AMS (MOS) subject classifications (1970). Primary 46B99; Secondary 46G05.

A separable Banach space with the Radon-Nikodým property that is not isomorphic to a subspace of a separable dual