We provide a partial answer to the question of Vladimir Kadets whether given an ℱ-basis of a Banach space X, with respect to some filter ℱ ⊂ 𝒫(ℕ), the coordinate functionals are continuous. The answer is positive if the character of ℱ is less than 𝔭. In this case every ℱ-basis is an M-basis with brackets which are determined by an element of ℱ.
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Let (Ω,𝓐,P) be a probability space and let τ: ℝ × Ω → ℝ be a mapping strictly increasing and continuous with respect to the first variable, and 𝓐-measurable with respect to the second variable. We discuss the problem of existence of probability distribution solutions of the general linear equation $F(x) = ∫_Ω F(τ(x,ω)) P(dω)$. We extend our uniqueness-type theorems obtained in Ann. Polon. Math. 95 (2009), 103-114.
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We show that any quasi-arithmetic mean $A_{φ}$ and any non-quasi-arithmetic mean M (reasonably regular) are inconsistent in the sense that the only solutions f of both equations $f(M(x,y)) = A_{φ}(f(x),f(y))$ and $f(A_{φ}(x,y)) = M(f(x),f(y))$ are the constant ones.
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Let (Ω,𝓐,P) be a probability space and let τ: ℝ × Ω → ℝ be strictly increasing and continuous with respect to the first variable, and 𝓐-measurable with respect to the second variable. We obtain a partial characterization and a uniqueness-type result for solutions of the general linear equation $F(x) = ∫_{Ω} F(τ(x,ω))P(dω)$ in the class of probability distribution functions.
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Let X be a Banach space. We study the circumstances under which there exists an uncountable set 𝓐 ⊂ X of unit vectors such that ||x-y|| > 1 for any distinct x,y ∈ 𝓐. We prove that such a set exists if X is quasi-reflexive and non-separable; if X is additionally super-reflexive then one can have ||x-y|| ≥ slant 1 + ε for some ε > 0 that depends only on X. If K is a non-metrisable compact, Hausdorff space, then the unit sphere of X = C(K) also contains such a subset; if moreover K is perfectly normal, then one can find such a set with cardinality equal to the density of X; this solves a problem left open by S. K. Mercourakis and G. Vassiliadis (2015).
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