A note on the 2-dual space of $L^p[0,1]$

TL;DR

This paper investigates the structure of 2-dual spaces of $L^p[0,1]$, providing explicit computations with different norms, filling a gap in the literature on dual spaces of function spaces.

Contribution

It offers explicit descriptions of 2-dual spaces of $L^p[0,1]$ using standard and alternative norms, extending known results from sequence spaces to function spaces.

Findings

Explicit computation of 2-dual spaces with $ orm{.}_p$ norm

Analysis of 2-dual spaces with Gähler and Gunawan norms

Extension of n-dual space concepts to $L^p[0,1]$

Abstract

In the present note, we are interested in bounded 2-functionals and 2-dual spaces of $L^p[0,1]$. The 2-dual spaces of the sequence space $l^p$ is considered in the literature. But interestingly an explicit computation of $Lp$ spaces has not been considered though n-duals of general normed spaces have been considered. We shall consider the 2-dual spaces with the usual $\|.\|_p$ norm and with respect to the G\"{a}hler and the Gunawan norm. The n-dual space of $L^p[0,1]$ can be treated in a similar manner.