On the subspace of the $L^p$ space, which is an annihilator of an element not belonging to the dual space

TL;DR

This paper investigates a specific subspace of L^p spaces that annihilates certain functions outside the dual space, showing its density and the limitations of representing certain linear functionals.

Contribution

It demonstrates the density of a subspace in L^p and reveals that some extensions of linear functionals cannot be represented as integrals against functions.

Findings

The subspace Y is dense in L^p(E).

Extensions of certain functionals cannot be represented as integrals.

The subspace annihilates functions not in the dual space.

Abstract

Let $E$ be a Lebesgue measurable subset of ${\mathbb R}^n$, $p\in [1,\infty)$. We consider the subspace $Y\subset L^p(E)$, which is an annihilator of the Lebesgue measurable ${{\cal L}^{n}}$-a.e. finite function $g$ that does not belong to the dual space of $L^p(E)$. It is shown that the subspace $Y$ is dense in $L^p(E)$. Moreover, the Hahn-Banach theorem's extension $\bar T_g\in [L^p(E)]^*$ of the bounded on $Y$ functional $h\mapsto \int_E g(x)h(x)\,dx$, $h\in Y$, can not be represented in the form $\bar T_g(h)= \int_E g(x)h(x)\,dx$, $h\in L^p(E)$.