Weak orthogonality
Anatole Beck, Peter Warren · Pacific Journal of Mathematics · 1972
Two Hilbert space-valued functions, / and g, are orthogonal iff [ μ(dω) = 0, where denotes the inner product of the Hilbert space.This paper concerns an analogous condition for Banach space-valued functions where, in general, no inner product structure is given.DEFINITION 1.1.Let (42, Σ, μ) be a measure space and let / be a function from Ω into a Banach space X.Let X* denote the dual of X. f is said to be almost separably-valued iff there exists a separable subspace X o c X such that μ{f~ι{XIX = Σ*U), u a{k) > a.e .For these ω, defineIt is easy to show, using the inequalities of Cauchy and Bessel, that \K(ω)\£\\f(ω)\\ \\g(ω)\\ .== 0 for all i , since the g n 's are orthonormal.