Drop property equals reflexivity. (English) Zbl 0652.46009

Montesinos, V.

Stud. Math. 87, 93-100 (1987).

This paper investigates some characterizations of the drop property for Banach spaces introduced by S. Rolewicz in Stud. Math. 85, 27-35 (1986; Zbl 0642.46011). The main results are:
i) Drop Property \(\Leftrightarrow\) ii) Property \(\alpha\) \(\Leftrightarrow\) iii) Property H of Radon-Riesz + reflexivity.
(ii) \(\Rightarrow\) i) \(\Rightarrow\) reflexivity was already proved by Rolewicz).
reflexivity \(\Leftrightarrow\) Drop property for an equivalent norm (this last result gives another proof of a characterization of the reflexivity found by D. Kutzarova)
weakly 2-rotundity \(\Rightarrow\) Drop Property
\(k\)-rotundity \(\Rightarrow\) Drop Property
(the converse is false). The author then gives some stability theorems about Drop Property and some examples, in particular non-super-reflexive spaces with Drop Property.

Reviewer: P.Mazet

MSC:

46B20	Geometry and structure of normed linear spaces
46B10	Duality and reflexivity in normed linear and Banach spaces

Keywords:

drop property for Banach spaces; stability theorems; non-super-reflexive spaces with Drop Property

Citations:

Zbl 0642.46011