Homeomorphism between close relatives of Hilbertian balls

Abstract: We present a solution to some problems posed by the author and Kalenda. We show that the closed ball of nonseparable Hilbert space in its weak topology is homeomorphic to its positive part, as well as to its product with the Hilbert cube. In the separable setting we obtain that there is a weak homeomorphism of the closed unit ball of $\ell_2$ onto its positive part that preserves the norm, and via a result of Dijkstra and van Mill, the same is true for the ball of $\ell_\infty=\ell_1^*$ in the weak$^*$ topology. All spaces $B(κ,a,b)$ considered by the author and Kalenda are shown to be homeomorphic. The solution has been found by AI (Chatgpt 5.5), the role of the author has been to ask the right questions, check and understand the answers, and adapt the writing to his personal human taste.

• The paper presents explicit norm-preserving homeomorphisms between band-truncated Hilbertian balls and the canonical Hilbert ball.
• It employs piecewise linear maps and a recursive partition lemma to robustly handle both separable and uncountable index cases.
• The results unify various weakly compact convex sets in Banach space theory, simplifying their classification and advancing future functional analysis research.

Homeomorphism Classification of Relatives of Hilbertian Balls

Overview

This paper addresses longstanding open problems concerning the topological classification of weakly compact convex subsets closely related to the unit ball of a Hilbert space. The author constructs explicit homeomorphisms between various "relatives" of the Hilbertian ball, specifically spaces of the form $B(\kappa, a, b)$ (band-terminated $L_1$-type balls), and establishes that they are all homeomorphic under natural nontriviality conditions. This result significantly extends prior work on the fine structure of weakly compact convex sets in Banach space theory.

Background and Technical Setting

The canonical object of study is the set

$$B(\kappa) = \{ x \in [-1,1]^\kappa : \sum_{i \in \kappa} |x_i| \le 1 \}$$

in the product topology, which is the closed unit ball of $\ell_1(\kappa)$ in the weak$^*$ topology, and also homeomorphic to unit balls in weak topologies of other $\ell_p(\kappa)$ spaces for $1 < p < \infty$. When $\kappa$ is infinite, these sets are nonmetrizable, and their topology is deeply connected to the combinatorics of the underlying index set and geometrical features of Banach spaces.

A more general class is given by

$$B(\kappa, a, b) = \left\{ x \in \prod_{i \in \kappa} [a(i), b(i)] : \sum_{i \in \kappa} |x_i| \leq 1 \right\}$$

where $a < b$ coordinatewise and a summability condition ensures infinite-dimensionality and nontriviality.

Prior to this work, it was known that certain modifications (such as taking finite powers or certain quotient spaces) often destroy homeomorphic equivalence, with delicate distinctions arising in the nonseparable case. However, the question of whether "simple" relatives—such as the positive part $L_1$0 or band-restricted balls $L_1$1—are homeomorphic to $L_1$2 had remained unsolved. The present paper fully resolves these questions.

Main Results

Homeomorphism Universality

The chief theorem asserts that for any infinite cardinal $L_1$3, and under mild nontriviality assumptions on the bands $L_1$4, the space $L_1$5 is homeomorphic to $L_1$6. Explicitly,

• For any $L_1$7 with $L_1$8 and $L_1$9, $$B(\kappa) = \{ x \in [-1,1]^\kappa : \sum_{i \in \kappa} |x_i| \le 1 \}$$0 is homeomorphic to $$B(\kappa) = \{ x \in [-1,1]^\kappa : \sum_{i \in \kappa} |x_i| \le 1 \}$$1.

As a corollary, widely studied sets such as the positive ball $$B(\kappa) = \{ x \in [-1,1]^\kappa : \sum_{i \in \kappa} |x_i| \le 1 \}$$2, the half-ball $$B(\kappa) = \{ x \in [-1,1]^\kappa : \sum_{i \in \kappa} |x_i| \le 1 \}$$3, all band-restricted balls $$B(\kappa) = \{ x \in [-1,1]^\kappa : \sum_{i \in \kappa} |x_i| \le 1 \}$$4 for $$B(\kappa) = \{ x \in [-1,1]^\kappa : \sum_{i \in \kappa} |x_i| \le 1 \}$$5, and their associated products with the Hilbert cube $$B(\kappa) = \{ x \in [-1,1]^\kappa : \sum_{i \in \kappa} |x_i| \le 1 \}$$6, are all homeomorphic. Further, products with the interval $$B(\kappa) = \{ x \in [-1,1]^\kappa : \sum_{i \in \kappa} |x_i| \le 1 \}$$7 preserve the homeomorphism class; that is, $$B(\kappa) = \{ x \in [-1,1]^\kappa : \sum_{i \in \kappa} |x_i| \le 1 \}$$8 is homeomorphic to $$B(\kappa) = \{ x \in [-1,1]^\kappa : \sum_{i \in \kappa} |x_i| \le 1 \}$$9.

Explicit Norm-Preserving Homeomorphisms in the Separable Case

The separable (countable) case receives a particularly constructive description. The author produces a continuous bijection $\ell_1(\kappa)$0 that satisfies the strong norm preservation property

$\ell_1(\kappa)$1

Additionally, via results of Dijkstra and van Mill, the weak$\ell_1(\kappa)$2 ball of $\ell_1(\kappa)$3 also enjoys such a norm-preserving homeomorphism onto its positive part.

The Uncountable Construction

The work extends these constructions to the uncountable case using a partition lemma, reducing the uncountable ball into countable blocks where the countable homeomorphisms can be glued together, leveraging properties of product and sum topologies in the Tychonoff setting.

Methodology

A technically sophisticated step in the construction is the definition of norm-preserving piecewise linear homeomorphisms in two dimensions, which serve as "local bricks" for the higher-dimensional and uncountable case. The planar mapping is analyzed carefully for continuity, invertibility, and uniform control (Lipschitz estimates). The recursive formulas used in the countable case yield strong control over the construction and guarantee that the global mapping remains a homeomorphism with the desired norm-preserving properties.

An elementary but nontrivial partitioning lemma is established to extend the argument to uncountable index sets. This enables the decomposition of the uncountable ball into countably infinite summands where the countable constructions apply, then recombined via product topology.

Numerical and Structural Implications

The results contradict earlier intuitions that minor band modifications on high-dimensional weakly compact balls might lead to non-homeomorphic spaces, known to occur in some cases for certain polyadic and fiber order structures. Here, the author demonstrates robustness of the homeomorphism class under a wide class of band truncations and related operations, in sharp contrast to the delicate behavior previously observed for finite powers and chain conditions.

No explicit numerical computations are present in this work due to its topological focus, but the construction is explicit and effective, providing strong classification tools for future research in Banach space topology.

Theoretical and Practical Implications

These findings decisively advance the classification of uniform Eberlein compacta and provide clarity on the topological structure of weakly compact convex sets arising in functional analysis. For set-theoretic topology and Banach space theory, this removes certain obstacles in understanding when compact convex sets in dual or weak topologies may meaningfully be distinguished.

Practically, homeomorphism equivalence allows generalization of properties known for the canonical Hilbert ball to a much broader class of convex sets, simplifying arguments in compactness, universality, and descriptive set theory in infinite-dimensional settings. The constructive nature of the proof yields a framework for developing algorithms and mappings in infinite-dimensional convex analysis with preserved norms and controlled topological behavior.

Future Directions

Possible extensions include more detailed analysis of automorphism groups of these spaces, understanding homeomorphism invariants beyond the class covered here, and exploring whether similar universality holds for more general compact convex sets beyond band-truncated $\ell_1(\kappa)$4-balls. Furthermore, implications could be sought for the descriptive set-theoretic classification of compacta in relation to Banach space smoothness and extensibility.

The constructive method also suggests potential applications in functional analysis algorithms dealing with weak or weak$\ell_1(\kappa)$5 topologies where explicit norm-preserving homeomorphic representations are advantageous.

Conclusion

The paper establishes that all nontrivial band-truncated balls $\ell_1(\kappa)$6, including the positive and half balls, are homeomorphic to the standard Hilbertian ball $\ell_1(\kappa)$7. Explicit norm-preserving homeomorphisms are constructed in the separable case, and the technique extends to the uncountable setting via systematic partitioning. This unifies a substantial class of weakly compact convex sets in Banach space theory under a single homeomorphism type and clarifies their topological structure in both separable and nonseparable cases.