A Renorming of Nonreflexive Banach Spaces

William J. Davis, William B. Johnson
1973 Proceedings of the American Mathematical Society  
Every nonreflexive Banach space can be equivalently renormed in such a way that it is not isometrically a conjugate space. Dixmier [1] asked: "If A'is isomorphic to a conjugate Banach space, is X isometric to a conjugate space?" Klee [5] gave a negative solution by giving an equivalent norm for lx under which that space is not isometrically a dual space. Here, we show that such a norm exists for every nonreflexive Banach space. The result is precise since, obviously, if X is reflexive, it is
more » ... reflexive, it is isometrically a conjugate space under any equivalent renorming. The prototype of our first lemma is the renorming theorem of Kadec ([2], [3]) and Klee [6]. The version we give may have some novelty since we do not assume separability of X. We feel that the proof we sketch here is somewhat more revealing than existing proofs of the Kadec-Klee
doi:10.2307/2039469 fatcat:pmeolwtftfbdhgvfn4tdwdpv5q