Almost all positive continuous linear functionals can be extended

2020/09/24 by de Bruyn, Josse van Dobben
#46A40 (Primary) #47B65 #47L07 #52A20 (Secondary) #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.2009.11844

Abstract

Let F be an ordered topological vector space (over ℝ) whose positive cone F₊ is weakly closed, and let E ⊆ F be a subspace. We prove that the set of positive continuous linear functionals on E that can be extended (positively and continuously) to F is weak-* dense in the topological dual wedge E₊'. Furthermore, we show that this result cannot be generalized to arbitrary positive operators, even in finite-dimensional spaces.

Related

• Extensions of positive operators and functionals
• Extension of Operators from Weak^*-closed Subspaces of ℓ₁
• Non-unital operator systems that are dual spaces
• The Kreps-Yan theorem for L^∞
• Continuous extension of functions from countable sets