On the subspace of the $L^p$ space, which is an annihilator of an element not belonging to 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 ${\mathcal{L}}^n$-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 ${\bar{T}}_g\in [L^p(E){]}^{\ast }$ 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 ${\bar{T}}_g(h)={\int }_{E}g(x)h(x)dx$, $h\in L^p(E)$.

arXiv commentthere is a simpler and more general solution