Banach spaces that have normal structure and are isomorphic to a Hilbert space

Abstract

We prove that given a Hilbert space ( E , | | ⋅ | | ) \left ( {E,|| \cdot ||} \right ) , and | ⋅ | | \cdot | a norm on E E such that for all x ∈ E x \in E , 1 / β | x | ⩽ ‖ x ‖ ⩽ | x | 1/\beta \left | x \right | \leqslant \left \| x \right \| \leqslant \left | x \right | for some β \beta , if 1 ⩽ β > 2 1 \leqslant \beta > \sqrt 2 , then ( E , | ⋅ | ) \left ( {E,| \cdot |} \right ) satisfies a convexity property from which normal structure follows.