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A Lifting Result for Locally Pseudo-Convex Subspaces of L₀

100%
Sánchez F.
Bulletin of the Polish Academy of Sciences. Mathematics
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2006
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tom 54
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nr 3
231-236
EN
It is shown that if F is a topological vector space containing a complete, locally pseudo-convex subspace E such that F/E = L₀ then E is complemented in F and so F = E⊕ L₀. This generalizes results by Kalton and Peck and Faber.
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Quasi-homomorphisms

100%
Sánchez F.
Fundamenta Mathematicae
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2003
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tom 178
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nr 3
255-270
EN
We study the stability of homomorphisms between topological (abelian) groups. Inspired by the "singular" case in the stability of Cauchy's equation and the technique of quasi-linear maps we introduce quasi-homomorphisms between topological groups, that is, maps ω: 𝒢 → ℋ such that ω(0) = 0 and ω(x+y) - ω(x) - ω(y) → 0 (in ℋ) as x,y → 0 in 𝒢. The basic question here is whether ω is approximable by a true homomorphism a in the sense that ω(x)-a(x) → 0 in ℋ as x → 0 in 𝒢. Our main result is that quasi-homomorphisms ω:𝒢 → ℋ are approximable in the following two cases: ∙ 𝒢 is a product of locally compact abelian groups and ℋ is either ℝ or the circle group 𝕋. ∙ 𝒢 is either ℝ or 𝕋 and ℋ is a Banach space. This is proved by adapting a classical procedure in the theory of twisted sums of Banach spaces. As an application, we show that every abelian extension of a quasi-Banach space by a Banach space is a topological vector space. This implies that most classical quasi-Banach spaces have only approximable (real-valued) quasi-additive functions.
3
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Banach space techniques underpinning a theory for nearly additive mappings

68%
Sánchez F., Castillo J.
Instytut Matematyczny Polskiej Akademii Nauk
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