Espaces de suites réelles complètement métrisables

• Autorzy: Pierre Casevitz
• Języki publikacji: FR EN

Abstrakty

EN

Let X be an hereditary subspace of the Polish space $ℝ^{ω}$ of real sequences, i.e. a subspace such that [x = (xₙ)ₙ ∈ X and ∀n, |yₙ| ≤ |xₙ|] ⇒ y = (yₙ)ₙ ∈ X. Does X admit a complete metric compatible with its vector structure? We have two results:
∙ If such an X has a complete metric δ, there exists a unique pair (E,F) of hereditary subspaces with E ⊆ X ⊆ F, (E,δ) complete separable, and F complete maximal in a strong sense. On E and F, the metrics have a simple form, and the spaces E are Borel (Π₃⁰ or Σ₂⁰) in $ℝ^{ω}$. In particular, if X is separable, then X = E.
∙ If X is an hereditary space, analytic as a subset of $ℝ^{ω}$, we can find a subspace of X strongly isomorphic to the space c₀₀ of finite sequences, or we can find a pair (E,F) and a metric with the same properties around X. If X is Σ₃⁰ in $ℝ^{ω}$, we get a complete trichotomy describing the possible topologies of X, which makes precise a result of [C], but for general X's, there are examples of various situations.

Kategorie tematyczne

• 06F20: Ordered abelian groups, Riesz groups, ordered linear spaces
• 54E52: Baire category, Baire spaces
• 46A19: Other ``topological'' linear spaces (convergence spaces, ranked spaces, spaces with a metric taking values in an ordered structure more general than 𝐑 , etc.)
• 03E15: Descriptive set theory
• 46A45: Sequence spaces (including K\"othe sequence spaces)
• 54D55: Sequential spaces

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Fundamenta Mathematicae
• Rocznik: 2001
• Tom: 168
• Numer: 3
• Strony: 199-235

Daty

wydano

2001

Twórcy

autor

Pierre Casevitz

• SDAD - Université de Caen, Campus II, Boulevard Maréchal Juin, 1, Esplanade de la Libération, BP 5186, F-14032 Caen Cedex, France

Identyfikatory

DOI

10.4064/fm168-3-1