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The splitting relation for Fréchet spaces over non-archimedean fields

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Published/Copyright: February 8, 2012
Forum Mathematicum
From the journal Forum Mathematicum Volume 26 Issue 3

Abstract.

A complete valued field (K,|·|) is non-archimedean if its valuation satisfies the strong triangle inequality|a+b|≤max{|a|,|b|} for all a,b∈K. We say that a pair (E,F) of Fréchet spaces over a non-archimedean field 𝕂 is splitting if for every Fréchet space G over 𝕂 and for every closed subspace D of G such that D is isomorphic to F and G/D is isomorphic to E, we can infer that the subspace D is complemented in G. In this paper we study when a pair (E,F) of Fréchet spaces of countable type over 𝕂 is splitting. In particular, we show that a pair (As(a),Ar(b)) of power series spaces over 𝕂 is splitting if and only if s=∞ or s=1 and the set Ma,b of all finite limit points of the double sequence (ap/bq)p,q∈ℕ is bounded.

Keywords: Splitting relation; non-archimedean Köthe spaces
MSC: 46S10; 46A04; 46A11; 46A45

Funding source: National Center of Science, Poland

Award Identifier / Grant number: N N201 605340

The author wishes to thank the referee for Lemma A and suggesting many improvements.

Received: 2011-8-24
Revised: 2012-1-22
Published Online: 2012-2-8
Published in Print: 2014-5-1

© 2014 by Walter de Gruyter Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. Smooth global solutions for the two-dimensional Euler Poisson system
  3. Primitive ideals in quantum Schubert cells: Dimension of the strata
  4. Group valued null sequences and metrizable non-Mackey groups
  5. The splitting relation for Fréchet spaces over non-archimedean fields
  6. Bloch–Kato pro-p groups and locally powerful groups
  7. On Lp resolvent estimates for Laplace–Beltrami operators on compact manifolds
  8. The Ext functor and self-sums
  9. Auslander–Reiten translations in monomorphism categories
  10. An elementary proof of the vanishing of the second cohomology of the Witt and Virasoro algebra with values in the adjoint module
  11. Unitals over composition algebras
  12. Separation of bi-harmonic differential operators on Riemannian manifolds
https://doi.org/10.1515/forum-2011-0089
Keywords for this article
Splitting relation; non-archimedean Köthe spaces

Articles in the same Issue

  1. Frontmatter
  2. Smooth global solutions for the two-dimensional Euler Poisson system
  3. Primitive ideals in quantum Schubert cells: Dimension of the strata
  4. Group valued null sequences and metrizable non-Mackey groups
  5. The splitting relation for Fréchet spaces over non-archimedean fields
  6. Bloch–Kato pro-p groups and locally powerful groups
  7. On Lp resolvent estimates for Laplace–Beltrami operators on compact manifolds
  8. The Ext functor and self-sums
  9. Auslander–Reiten translations in monomorphism categories
  10. An elementary proof of the vanishing of the second cohomology of the Witt and Virasoro algebra with values in the adjoint module
  11. Unitals over composition algebras
  12. Separation of bi-harmonic differential operators on Riemannian manifolds
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