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A note on trees in conjugate Banach spaces. (English) Zbl 0537.46025
In this paper a simple proof is given of the following result due to C. Stegall [Trans. Am. Math. Soc. 206, 213-223 (1975; Zbl 0318.46056)]: if a Banach space X contains a separable subspace Y with non-separable conjugate $$Y^*$$, then for each $$\epsilon$$ such that $$0<\epsilon<1$$, the unit ball of $$X^*$$ contains an infinite $$\epsilon$$ /2-tree.
This proof also yields another theorem by C. Stegall [Isr. J. Math. 29, 408-412 (1978; Zbl 0374.46015)], namely, a Banach space X is an Asplund space if and only if $$X^*$$ has the RNP, without resorting to the construction in Stegall’s 1975 paper.

MSC:
 46B20 Geometry and structure of normed linear spaces 46B22 Radon-Nikodým, Kreĭn-Milman and related properties
Keywords:
tree; Asplund space