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تسجيل مستخدم جديدThe geometry of two-valued subsets of $L_{p}$-spaces
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تأليف
Anthony Weston
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Let $mathcal{M}(Omega, mu)$ denote the algebra of all scalar-valued measurable functions on a measure space $(Omega, mu)$. Let $B subset mathcal{M}(Omega, mu)$ be a set of finitely supported measurable functions such that the essential range of each $f in B$ is a subset of ${ 0,1 }$. The main result of this paper shows that for any $p in (0, infty)$, $B$ has strict $p$-negative type when viewed as a metric subspace of $L_{p}(Omega, mu)$ if and only if $B$ is an affinely independent subset of $mathcal{M}(Omega, mu)$ (when $mathcal{M}(Omega, mu)$ is considered as a real vector space). It follows that every two-valued (Schauder) basis of $L_{p}(Omega, mu)$ has strict $p$-negative type. For instance, for each $p in (0, infty)$, the system of Walsh functions in $L_{p}[0,1]$ is seen to have strict $p$-negative type. The techniques developed in this paper also provide a systematic way to construct, for any $p in (2, infty)$, subsets of $L_{p}(Omega, mu)$ that have $p$-negative type but not $q$-negative type for any $q > p$. Such sets preclude the existence of certain types of isometry into $L_{p}$-spaces.
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