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arXiv · 1802.08347

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

Dmitrii Prokhorov
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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 $\bar T_g(h)= \int_E g(x)h(x)\,dx$, $h\in L^p(E)$.

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Dmitrii Prokhorov. 2018-02-22. On the subspace of the $L^p$ space, which is an annihilator of an element not belonging to the dual space. https://arxiv.org/abs/1802.08347

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