The geometry of two-valued subsets of $L_p$-spaces

Rok, strany: 2017, 751 - 758

Let $\mathcal{M}(Ω, μ)$ denote the algebra of all scalar-valued measurable functions on a measure space $(Ω, μ)$. Let $B \subset \mathcal{M}(Ω, μ)$ 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 $p \in (0, ∞)$, $B$ has strict $p$-negative type when viewed as a metric subspace of $L_p(Ω, μ)$ if and only if $B$ is an affinely independent subset of $\mathcal{M}(Ω, μ)$ (when $\mathcal{M}(Ω, μ)$ is considered as a real vector space). It follows that every two-valued (Schauder) basis of $L_p(Ω, μ)$ has strict $p$-negative type. For instance, for each $p \in (0, ∞)$, 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 $p \in (2, ∞)$, subsets of $L_p(Ω, μ)$ 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.

ISO 690:

Weston, A. 2017. The geometry of two-valued subsets of $L_p$-spaces. In Mathematica Slovaca, vol. 67, no.3, pp. 751-758. 0139-9918. DOI: https://doi.org/10.1515/ms-2017-0007

APA:

Weston, A. (2017). The geometry of two-valued subsets of $L_p$-spaces. Mathematica Slovaca, 67(3), 751-758. 0139-9918. DOI: https://doi.org/10.1515/ms-2017-0007