Tusi Mathematical Research Group (TMRG)Advances in Operator Theory2538-225X3220180401Certain geometric structures of $Lambda$-sequence spaces4334505341210.15352/aot.1705-1164ENAtanu MannaIndian Institute of Carpet Technology, Chauri road, Bhadohi-221401, Uttar Pradesh, India.Journal Article20170515The $Lambda$-sequence spaces $Lambda_p$ for $1< pleqinfty$ and their generalized forms $Lambda_{hat{p}}$ for $1<hat{p}<infty$, $hat{p}=(p_n)$, $nin mathbb{N}_0$ are introduced. The James constants and strong $n$-th James constants of $Lambda_p$ for $1<pleqinfty$ are determined. It is proved that the generalized $Lambda$-sequence space $Lambda_{hat{p}}$ is a closed subspace of the Nakano sequence space $l_{hat{p}}(mathbb{R}^{n+1})$ of finite dimensional Euclidean space $mathbb{R}^{n+1}$, $nin mathbb{N}_0$. Hence it follows that sequence spaces $Lambda_p$ and $Lambda_{hat{p}}$ possess the uniform Opial property, $(beta)$-property of Rolewicz, and weak uniform normal structure. Moreover, it is established that $Lambda_{hat{p}}$ possesses the coordinate wise uniform Kadec--Klee property. Further, necessary and sufficient condition for element $xin S(Lambda_{hat{p}})$ to be an extreme point of $B(Lambda_{hat{p}})$ are derived. Finally, estimation of von Neumann-Jordan and James constants of two dimensional $Lambda$-sequence space $Lambda_2^{(2)}$ are carried out. Upper bound for the Hausdorff matrix operator norm on the non-absolute type $Lambda$-sequence spaces is also obtained.http://www.aot-math.org/article_53412_20d5ffb6fef2fbf6e9bc4e0d2d893f7e.pdf