Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

An extension result for $(LB)$-spaces and the surjectivity of tensorized mappings

Andreas Debrouwere, Lenny Neyt

Abstract

We study an extension problem for continuous linear maps in the setting of $(LB)$-spaces. More precisely, we characterize the pairs $(E,Z)$, where $E$ is a locally complete space with a fundamental sequence of bounded sets and $Z$ is an $(LB)$-space, such that for every exact sequence of $(LB)$-spaces $$ 0 \rightarrow X \xrightarrowι Y \rightarrow Z \rightarrow 0$$ the map $$ L(Y,E) \to L(X, E), ~ T \mapsto T \circ ι$$ is surjective, meaning that each continuous linear map $X \to E$ can be extended to a continuous linear map $Y \to E$ via $ι$, under some mild conditions on $E$ or $Z$ (e.g. one of them is nuclear). We use our extension result to obtain sufficient conditions for the surjectivity of tensorized maps between Fréchet-Schwartz spaces. As an application of the latter, we study vector-valued Eidelheit type problems. Our work is inspired by and extends results of Vogt [24].

An extension result for $(LB)$-spaces and the surjectivity of tensorized mappings

Abstract

We study an extension problem for continuous linear maps in the setting of -spaces. More precisely, we characterize the pairs , where is a locally complete space with a fundamental sequence of bounded sets and is an -space, such that for every exact sequence of -spaces the map is surjective, meaning that each continuous linear map can be extended to a continuous linear map via , under some mild conditions on or (e.g. one of them is nuclear). We use our extension result to obtain sufficient conditions for the surjectivity of tensorized maps between Fréchet-Schwartz spaces. As an application of the latter, we study vector-valued Eidelheit type problems. Our work is inspired by and extends results of Vogt [24].
Paper Structure (15 sections, 30 theorems, 107 equations)

This paper contains 15 sections, 30 theorems, 107 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. Inductive spectra
  5. Projective spectra
  6. Sequence spaces
  7. An extension result for $(LB)$-spaces
  8. The conditions $(\mathop{\mathrm{S}}\nolimits)$ and $(\mathop{\mathrm{WS}}\nolimits)$
  9. Statement of the main result
  10. Weak $E$-acyclicity and the derived projective limit
  11. A characterization of $\mathop{\mathrm{Proj}}\nolimits^{1} \mathscr{L}(Z, E) = 0$
  12. The conditions $(\mathop{\mathrm{A}}\nolimits)$ and $(\mathop{\mathrm{D\Omega}}\nolimits)$
  13. The splitting of exact sequences of $(LB)$-spaces
  14. Surjectivity of tensorized mappings
  15. Vector-valued Eidelheit families

Key Result

Proposition 2.1

W-DerivFunctFuncAnal Let $\mathscr{Y} = (Y_{n}, \varrho_{m}^{n})$ and $\mathscr{Z} = (Z_{n}, \tau_{m}^{n})$ be two projective spectra and let $f = (f_{n})_{n \in \mathbb{N}} : \mathscr{Y} \rightarrow \mathscr{Z}$ be a morphism. Suppose that for all $n \in \mathbb{N}$ there is $m \in \mathbb{N}$ such

Theorems & Definitions (69)

  • Proposition 2.1
  • Proposition 2.2
  • proof
  • Definition 3.1
  • Proposition 3.2
  • proof
  • Corollary 3.3
  • proof
  • Corollary 3.4
  • proof
  • ...and 59 more