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On weak*-basic sequences in duals and biduals of spaces C(X) and Quojections

Jerzy Kakol, Manuel Lopez-Pellicer, Wieslaw Sliwa

TL;DR

The paper investigates the existence of $w^{*}$-basic sequences in the duals and biduals of natural function spaces, notably $C_k(X)$ and $C_k(X\times Y)$, for infinite Tychonoff spaces. It establishes that $C_k(X\times Y)^*$ contains a $w^{*}$-basic sequence and that $C_k(X)^{**}$ likewise contains one, with a concrete compact-case construction of a sequence $(\mu_n)$ whose subsequences yield strongly normal $w^{*}$-basic subsequences in $C(X\times Y)^*$. Extending beyond Banach spaces, the work shows that for every quojection $E$, the bidual $E^{**}$ has a $w^{*}$-basic sequence, and it discusses inductive limits of Fréchet spaces (LF-spaces), where duals similarly admit $w^{*}$-basic sequences. The paper also highlights open problems, including whether the dual of every infinite-dimensional Banach space contains a $w^{*}$-basic sequence, and presents various examples and questions for $C(X)$ spaces and inductive limits.

Abstract

We show that for infinite Tychonoff spaces X and Y the weak*-dual of Ck(X x Y) contains a basic sequence; moreover, the weak*-bidual of Ck(X) contains such a sequence as well. When X and Y are infinite compact spaces, we single out a concrete sequence (μn) of finitely supported signed measures on X x Y with quantitative small-rectangle estimates, and we prove that every subsequence of (μn) admits a further subsequence which is strongly normal and forms a weak*-basic sequence in the dual C(X x Y)* of the Banach space C(X x Y). We also study the weak*-basic sequence problem for Frechet locally convex spaces in the class of quojections, and prove that for every quojection E the bidual E** admits a weak*-basic sequence, while a long-standing open problem asks whether the dual of every infinite-dimensional Banach space admits a basic sequence in the weak*-topology. Several examples and open questions are included, in particular for spaces C(X) and for inductive limits of Frechet spaces.

On weak*-basic sequences in duals and biduals of spaces C(X) and Quojections

TL;DR

The paper investigates the existence of -basic sequences in the duals and biduals of natural function spaces, notably and , for infinite Tychonoff spaces. It establishes that contains a -basic sequence and that likewise contains one, with a concrete compact-case construction of a sequence whose subsequences yield strongly normal -basic subsequences in . Extending beyond Banach spaces, the work shows that for every quojection , the bidual has a -basic sequence, and it discusses inductive limits of Fréchet spaces (LF-spaces), where duals similarly admit -basic sequences. The paper also highlights open problems, including whether the dual of every infinite-dimensional Banach space contains a -basic sequence, and presents various examples and questions for spaces and inductive limits.

Abstract

We show that for infinite Tychonoff spaces X and Y the weak*-dual of Ck(X x Y) contains a basic sequence; moreover, the weak*-bidual of Ck(X) contains such a sequence as well. When X and Y are infinite compact spaces, we single out a concrete sequence (μn) of finitely supported signed measures on X x Y with quantitative small-rectangle estimates, and we prove that every subsequence of (μn) admits a further subsequence which is strongly normal and forms a weak*-basic sequence in the dual C(X x Y)* of the Banach space C(X x Y). We also study the weak*-basic sequence problem for Frechet locally convex spaces in the class of quojections, and prove that for every quojection E the bidual E** admits a weak*-basic sequence, while a long-standing open problem asks whether the dual of every infinite-dimensional Banach space admits a basic sequence in the weak*-topology. Several examples and open questions are included, in particular for spaces C(X) and for inductive limits of Frechet spaces.
Paper Structure (4 sections, 17 theorems, 13 equations)

This paper contains 4 sections, 17 theorems, 13 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proof of Theorem \ref{['VVI']} and more on spaces $C_k(X)$, $C_p(X)$
  4. Proof of Theorem \ref{['VI']}

Key Result

Theorem 1

For a lcs $E$ consider the following statements: Then $(2)\Rightarrow(3)\Rightarrow(4)\Leftrightarrow(1)$. Moreover, all statements are equivalent if $E$ is a Fréchet lcs.

Theorems & Definitions (32)

  • Theorem 1: Śliwa--Wójtowicz
  • Theorem 2
  • Theorem 3: Ka̧kol-Śliwa
  • Remark 1
  • Theorem 4: Argyros--Dodos--Kanellopoulos
  • Corollary 5
  • Theorem 6
  • Remark 2
  • Remark 3
  • Theorem 9: Śliwa
  • ...and 22 more