The $\Lambda$-sequence spaces $\Lambda_p$ for $1< p\leq\infty$ and its generalization $\Lambda_{ \hat{ p}}$ for $1<\hat{ p}<\infty$, $\hat{ p}=(p_n)$ is introduced. The James constants and strong $n$-th James constants of $\Lambda_p$ for $1<p\leq\infty$ is determined. It is proved that generalized $\Lambda$-sequence space $\Lambda_{ \hat{ p}}$ is embedded isometrically in the Nakano sequence space $l_{ \hat{ p}}(\mathbb{ R}^{ n+1})$ of finite dimensional Euclidean space $\mathbb{ R}^{ n+1}$. Hence it follows that sequence spaces $\Lambda_p$ and $\Lambda_{ \hat{ p}}$ possesses the uniform Opial property, property $(\beta)$ 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 conditions for element $x\in S(\Lambda_{ \hat{ p}})$ 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 $\Lambda_2^{ (2)}$ is being carried out.
Categories: math.FA