关于$L^p$空间的非对偶空间的零化子
On the subspace of the $L^p$ space, which is an annihilator of an element not belonging to the dual space
摘要：
设$E$是${\mathbb R}^n$的Lebesgue可测子集，$p\in [1,\infty)$。我们考虑子空间$Y\subset L^p(E)$，它是Lebesgue可测${{\cal L}^{n}}$ -a.e.的零化子.不属于$L^p(E)$的对偶空间的有限函数$g$。证明了$L^p(E)$中的子空间$Y$是稠密的。此外，Hahn-Banach定理对$Y$上有界泛函$h\mapsto \int_E g(x)h(x)\,dx$的推广$\bar T_g\in [L^p(E)]^*$，$h\in Y$不能表示为$\bar T_g(h)= \int_E g(x)h(x)\,dx$，$h\in L^p(E)$ .
英文摘要：
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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