Abstract
We prove a uniform boundedness principle for the Lipschitz seminorm of continuous, monotone, positively homogeneous, and subadditive mappings on suitable cones of functions. The result is applicable to several classes of classically nonlinear operators.
Highlights
Introduction and preliminariesUniform boundedness principle for bounded linear operators (Banach-Steinhaus theorem) in one of the cornerstones of classical functional analysis
In this article we prove a new uniform boundedness principle for monotone, positively homogeneous, subadditive and Lipschitz mappings defined on a suitable cone of functions (Theorem 2.2)
This result is applicable to several classes of classically non-linear operators (Examples 2.4, 2.5, and Remarks 2.7, 2.8)
Summary
Introduction and preliminariesUniform boundedness principle for bounded linear operators (Banach-Steinhaus theorem) in one of the cornerstones of classical functional analysis (see e.g [12], [1], [14] and the references cited there). In this article we prove a new uniform boundedness principle for monotone, positively homogeneous, subadditive and Lipschitz mappings defined on a suitable cone of functions (Theorem 2.2). Let Y ⊂ X be a vector space and let Y+ denote the positive cone of Y , i.e., the set of all φ ∈ Y such that φ(ω) ≥ 0 for all (almost all) ω ∈ Ω.
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