Local uniform Kadec-Klee property (LUKK) and modulus of (LUKK) (Q2198775)

Summary: 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 (\textit{LUKK}) and local near uniform convexity (\textit{LNUC}), uniformly Kadec-Klee (\textit{UKK}), (\textit{H}) are investigated in Banach space. Furthermore, the modulus \(P_X^L(\varepsilon)\) of (\textit{LUKK}) and the modulus \(\Delta_X^L (\varepsilon)\) of (\textit{LNUC}) are introduced and the relationship of size between \(P_X^L (\varepsilon)\) and \(\Delta_X^L (\varepsilon)\) is also investigated in Banach space. Finally, several formulas for \(P_X^L (\varepsilon)\) are calculated in classical Banach space \(l^p\).