On the second dual of the Lorentz space

Abstract:

If $\phi (t) = {t^{1/p}}(p > 1)$ and $(X,\mathcal {S},\mu )$ is a completely nonatomic finite measure space, then the dual of the Lorentz space ${N_\phi }$ is denoted by ${M_\phi }$ and the closure of the simple functions in ${M_\phi }$ by $M_\phi ^0$. It is known that ${(M_\phi ^0)^ * } = {N_\phi }$. In this note we show that given a positive number $\beta < 1$ it is possible to construct a set of contractive embeddings of $({l_\infty }/{c_0})$ into ${({M_\phi }/M_\phi ^0)^ * }$, each of which is bounded below by $M = M(\beta ) \to 1\;{\text {as}}\;\beta \to {{\text {0}}^ + }$. The union of the ranges of these embeddings is a total set in ${({M_\phi }/M_\phi ^0)^ * }$.