Abstract
A new geometry property and two new moduli are introduced in Banach space. First, the concept of local uniform Kadec-Klee property (LUKK) is introduced and the implication relationships between LUKK and local near uniform convexity LNUC, uniformly Kadec-Klee (UKK), (H) are investigated in Banach space. Furthermore, the modulus PXLε of (LUKK) and the modulus ΔXLε of LNUC are introduced and the relationship of size between PXLε and ΔXLε is also investigated in Banach space. Finally, several formulas for PXLε are calculated in classical Banach space lp.
Highlights
Let ðX, k·kÞ be a Banach space, X∗ be the dual space of X
We introduce a new geometric property ðLUKKÞ that lies between two classical geometric properties (UKK) and (H)
Two new moduli PLXðεÞ and ΔLXðεÞ for (LUKK) and ðLNUCÞ are introduced in Banach spaces; these new notions introduced in our paper play a very significant role in some recent trends of the geometric theory of Banach spaces
Summary
Let ðX, k·kÞ be a Banach space, X∗ be the dual space of X. The functions ΔXðεÞ and Δb XðεÞ are called the moduli of noncompact convexity with Hausdorff measure and Kuratowski measure of X, respectively. PXðεÞ = inf f1−∥x∥ : ∃x ∈ UðXÞand fxng ⊂ UðXÞs:t:xn ⟶ w x and sepðxnÞ ≥ εg, ð4Þ where sepðxnÞ = inf f∥xn − xm∥ : m ≠ ng He proved that X has ðUKKÞ property if and only if PXðεÞ > 0 whenever ε ∈ 1⁄20, 1. Normal structure and moduli of ðUKKÞ, ðNUCÞ, and ðUKKÞ∗ in Banach spaces have been deeply investigated by Satit Saejung and Ji Gao. The new kind of Banach spaces: ðsemi − UKKÞ, ðsemi − NUCÞ, modulus of ðsemi − UKKÞ, and modulus of ðsemi − NUCÞ are introduced in terms of this u-separation measure in their paper.(see [12]). An effective method is to introduce new geometric properties for Banach space and to define an appropriate function, usually called a modulus or a geometric constant. Partington and the modulus of noncompact convexity with Hausdorff measure ðΔXðεÞÞ obtain two new moduli PLXðεÞ and ΔLXðεÞ, and we observe that ΔLXðεÞ ≥ PLXðεÞ
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