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

TL;DR

This paper investigates the existence of weak*-basic sequences in duals and biduals of spaces C(X) and quojections, providing concrete examples and addressing longstanding open problems in functional analysis.

Contribution

It demonstrates the presence of weak*-basic sequences in duals and biduals of C(X) and quojections, and introduces concrete sequences with quantitative estimates, advancing understanding of the weak*-basic sequence problem.

Findings

Weak*-basic sequences exist in duals and biduals of C(X) and quojections.

Constructed concrete sequences with small-rectangle estimates in C(X x Y).

Open problems regarding basic sequences in duals of Banach spaces are discussed.

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 ({\mu}n) of finitely supported signed measures on X x Y with quantitative small-rectangle estimates, and we prove that every subsequence of ({\mu}n) 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.