Construction of standard exact sequences of power series spaces

Volume 112 / 1995

Markus Poppenberg, Studia Mathematica 112 (1995), 229-241 DOI: 10.4064/sm-112-3-229-241


The following result is proved: Let $Λ_R^p(α)$ denote a power series space of infinite or of finite type, and equip $Λ_R^p(α)$ with its canonical fundamental system of norms, R ∈ {0,∞}, 1 ≤ p < ∞. Then a tamely exact sequence (⁎) $0 → Λ_{R}^{p}(α) → Λ_{R}^{p}(α) → Λ_{R}^{p}(α)^ℕ → 0$ exists iff α is strongly stable, i.e. $lim_n α_{2n}/α_n = 1$, and a linear-tamely exact sequence (*) exists iff α is uniformly stable, i.e. there is A such that $lim sup_n α_{Kn}/α_n ≤ A < ∞$ for all K. This result extends a theorem of Vogt and Wagner which states that a topologically exact sequence (*) exists iff α is stable, i.e. $sup_n α_{2n}/α_n < ∞$.


  • Markus Poppenberg

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