{"paper":{"title":"On the subspace of the $L^p$ space, which is an annihilator of an element not belonging to the dual space","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Dmitrii Prokhorov","submitted_at":"2018-02-22T23:38:35Z","abstract_excerpt":"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 $\\bar T_g(h)= \\int_E g(x)h(x)\\,dx$, $h\\in L^p(E)$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1802.08347","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}