Details

Autor(en) / Beteiligte

• Shang, Shaoqiang

Titel

The Ball-Covering Property on Dual Spaces and Banach Sequence Spaces

Ist Teil von

• Acta mathematica scientia, 2021-03, Vol.41 (2), p.461-474

Ort / Verlag

Singapore: Springer Singapore

Erscheinungsjahr

2021

Quelle

Alma/SFX Local Collection

Beschreibungen/Notizen

• In this paper, we prove that ( X, p ) is separable if and only if there exists a w *-lower semicontinuous norm sequence { p n } n = 1 ∞ of ( X*, p ) such that (1) there exists a dense subset G n of X * such that p n is Gâteaux differentiable on G n and dp n ( G n ) ⊂ X for all n ∊ N ; (2) p n ≤ p and p n → p uniformly on each bounded subset of X *; (3) for any α ∈ (0, 1), there exists a ball-covering { B ( x i n * r i n ) } i = 1 ∞ of ( X *, p n ) such that it is α -off the origin and x i, n * ∈ G n . Moreover, we also prove that if X i is a Gâteaux differentiability space, then there exist a real number α > 0 and a ball-covering B i of X i such that B i is α -off the origin if and only if there exist a real number α > 0 and a ball-covering B of l ∞ ( X i ) such that B is α -off the origin.