Weak*-Basic Sequence Problem

• Weak*-Basic Sequence Problem is defined as determining whether every dual of an infinite-dimensional Banach or locally convex space contains a weak*-basic (Schauder) sequence.
• Recent advances employ techniques like small-rectangle measures, strongly normal sequences, and structural analysis of quojections and LF-spaces to extract such sequences.
• The problem remains open in full generality, prompting further research into the conditions under which C-spaces and biduals admit weak*-basic sequences.

The Weak*-Basic Sequence Problem concerns the existence of Schauder basic sequences in the dual space of infinite-dimensional Banach and locally convex spaces equipped with the weak*-topology. A sequence in a dual space is weak*-basic if its linear span is weak*-closed and the canonical coordinate projections are weak*-continuous. The problem remains open in full generality but has seen substantial progress, especially for specific function spaces and classes of locally convex spaces such as quojections. Key recent advances involve the duals and biduals of spaces of continuous functions, as well as the extraction of weak*-basic sequences via strongly normal sequences of measures and structural analysis of inductive limits and projective limits in topological vector spaces.

1. Definitions and Fundamental Concepts

Let $E$ be a locally convex space with continuous dual $E^*$. The weak topology on $E$ is denoted $\sigma(E, E^*)$, while the weak*-topology on $E^*$, denoted $w^* = \sigma(E^*, E)$, is the topology of pointwise convergence on $E$. The strong topology on $E^*$ is $\beta(E^*, E)$.

A sequence $(e_n) \subset E$ is a Schauder basis if every $x \in E$ can be uniquely expressed as $x = \sum_{n=1}^\infty b_n e_n$, converging in the topology of $E$, with continuous coefficient functionals. A basic sequence is a Schauder basis for its closed linear span. A weak*-basic sequence in $E^*$ is a basic sequence in $(E^*, w^*)$, i.e., its span is weak*-closed and the coordinate projections are weak*-continuous.

A Fréchet space $E$ is a quojection if every quotient by the kernel of a continuous seminorm is a Banach space; equivalently, such spaces are strict projective limits of Banach spaces with surjective bonding maps (Kakol et al., 27 Jan 2026).

2. The Classical and Weak*-Basic Sequence Problem

The central open question is whether the dual $E^*$ of any infinite-dimensional Banach space $E$ necessarily contains a weak*-basic sequence. Specifically, does every such dual $(E^*, w^*)$ admit a countably infinite sequence $(e_n^*)$ serving as a Schauder basic sequence? Affirmative answers are known in special cases, but the general case remains unresolved (Kakol et al., 27 Jan 2026).

3. Main Positive Results for $C$-Spaces and Quojections

Significant recent progress has focused on spaces of continuous functions and classes of locally convex spaces:

• For infinite Tychonoff spaces $X, Y$, the dual of $C_k(X \times Y)$ with the compact-open topology contains a weak*-basic sequence. The bidual $C_k(X)^{**}$ (again with its weak*-topology) contains a weak*-basic sequence.
• If $C_k(X)$ is barrelled (e.g., $X$ metrizable), its dual already has a weak*-basic sequence.
• If $X$ contains an infinite discrete $C^*$-embedded subset, then $C_k(X)^*$ contains a weak*-basic sequence.
• For infinite compact $X, Y$, a concrete sequence $(\mu_n)$ of signed measures on $X \times Y$ is constructed with small-rectangle estimates, guaranteeing that every subsequence contains a further strongly normal subsequence, which in turn has a weak*-basic subsequence in $C(X \times Y)^*$. Explicitly, for each $n$, finite sets $K_n \subset X$ with $|K_n| = 2^n$ and $L_n \subset Y$ with $|L_n| = n$ are selected, along with a bijection $\varphi_n: K_n \to \{-1,1\}^{L_n}$, and the measure

$$\mu_n = \frac{1}{n 2^n} \sum_{(s,j) \in K_n \times L_n} \varphi_n(s)(j) \delta_{(s,j)}$$

satisfies $\|\mu_n\| = 1$ and small-rectangle estimates of order $1/\sqrt{n}$.

These results extend to biduals of Banach spaces and biduals of quojections: for any infinite-dimensional Banach space $E$, the bidual $E^{**}$ always contains a weak*-basic sequence, as does $E^{**}$ when $E$ is a quojection or a strict inductive limit (LF-space) of Banach spaces (Kakol et al., 27 Jan 2026).

4. Construction Techniques and Key Structural Arguments

The construction of weak*-basic sequences draws on both measure-theoretic and topological techniques:

• Small-Rectangle Measures and Strong Normality: The sequence $(\mu_n)$ defined above has support on grids $K_n \times L_n$ and satisfies norm and rectangle estimates:

$$\frac{1}{2\sqrt{\pi}} \frac{1}{\sqrt{n}} < \sup_{A \times B \subset X \times Y} |\mu_n(A \times B)| < \frac{2}{\sqrt{\pi}} \frac{1}{\sqrt{n}}$$

and for all elementary tensors $f \otimes g$,

$$|\mu_n(f \otimes g)| \le \frac{8}{\sqrt{\pi}}\frac{1}{\sqrt{n}}\|f \otimes g\|_\infty.$$

For any subsequence, a further sub-subsequence ensures summability of these estimates, yielding strong normality in the sense of Śliwa [J. Math. Soc. Japan 64 (2012), 387–397].

• Extraction of Weak*-Basic Subsequence: Every strongly normal sequence in the dual of a Banach space contains a weak*-basic subsequence per Śliwa's result.
• Structural Methods with Quojections and LF-spaces: The strong dual of a quojection is a strict (LB)-space. By Saxon–Moscatelli, strict inductive limits of Banach spaces contain complemented copies of $\varphi \cong \mathbb{R}^{\mathbb{N}}$ (product topology). Passing to the dual, $$(\varphi^*, w^*) \cong (\mathbb{R}^{\mathbb{N}}, \mathrm{product})$$, which has the standard unit vector basis as a Schauder basis. Thus, $E^{**}$ contains a complemented weak*-basic sequence (Kakol et al., 27 Jan 2026).

5. Examples, Special Cases, and Counterexamples

• For non-pseudocompact $X$, $C_p(X)$ and $C_k(X)$ each contain complemented copies of $\mathbb{R}^{\mathbb{N}}$, ensuring their weak*-duals have basic sequences.
• If $X$ is pseudocompact but contains an infinite discrete $C^*$-embedded subset, $C_k(X)$ still has a separable quotient admitting a basis, hence yields a weak*-basic sequence.
• There exist exotic barrelled spaces with no infinite-dimensional separable quotient; for such spaces, the dual $E^*$ admits no weak*-basic sequence. This demonstrates non-universality and subtlety of the problem in the barrelled non-separable case (Kakol et al., 27 Jan 2026).

6. Open Problems and Directions

Several significant questions remain unresolved:

Problem Code	Problem Statement
IV	Give a proof of existence of weak*-basic sequence in $E^{**}$ (for Banach $E$) avoiding reliance on the Argyros–Dodos–Kanellopoulos separable-quotient theorem.
Prob Char	Characterize locally convex spaces $E$ whose dual $E^*$ admits a weak*-basic sequence.
WU	Does $C_p(X)^*$ always contain a weak*-basic sequence for infinite compact $X$?
su	Does $C_k(X)^*$ always contain a weak*-basic sequence for infinite Tychonoff $X$?
RR	If $C_k(X)$ (or $C_p(X)$) has an infinite-dimensional separable quotient, must one exist with a Schauder basis?
LF-dual	Does the dual of every (strict) (LF)-space admit a weak*-basic sequence?

A plausible implication is that further progress may depend upon refined structure theory for locally convex spaces, the development of new extraction principles for basic sequences, or a characterization of "obstruction spaces" excluding the existence of weak*-basic sequences (Kakol et al., 27 Jan 2026).