Pointwise function spaces over compacta are not weak Banach spaces

Abstract: Let $K$ be a compact Hausdorff space and let $E$ be an infinite-dimensional real Banach space. We prove that there is no continuous bijection $h\colon C_p(K)\to E_w$ whose inverse is continuous at $h(0)$. Consequently, $C_p(K)$ and $C_w(L)$ are not homeomorphic for any infinite compact Hausdorff spaces $K$ and $L$. This settles Krupski's problem and its two-space version due to Krupski and Marciszewski, and answers a question of Kąkol, Leiderman, and Michalak concerning $C_p([0,1])$ and weak Banach spaces.

• The paper proves that for every compact Hausdorff space K and infinite-dimensional Banach space E, no continuous bijection C_p(K) → E_w can have an inverse continuous at the image of zero.
• The authors combine a linearly independent norm-compact arc in E*, a Baire category argument on finite supports, and a pointwise-null cut-off lemma to derive a contradiction.
• The result rules out homeomorphisms C_p(K) ≅ C_w(L) for all infinite compacta, while leaving the general Tychonoff case and bijections without any inverse-continuity assumption open.

The problem and its history

For a Tychonoff space $X$, $C_p(X)$ denotes the space of continuous real-valued functions with the topology of pointwise convergence, while for a Banach space $E$, $E_w$ denotes $E$ with its weak topology. Whether a pointwise function space over an infinite compactum can be homeomorphic to a weak Banach space is a question posed by M. Krupski in 2016, sharpened to a two-space version by Krupski and Marciszewski, and extended by Kąkol, Leiderman, and Michalak, who asked specifically whether $C_p([0,1])$ can be homeomorphic to $E_w$ for a separable Banach space $E$. Earlier work settled special cases: compact metrisable $C$-spaces [Krupski 2016], finite-dimensional Valdivia compacta [Krupski–Marciszewski 2017], and scattered $K$ via the Fréchet–Urysohn dichotomy. Kąkol, Leiderman, and Michalak also showed that any homeomorphism $C_p(X)$0 forces $C_p(X)$1 to be a countable union of compacta, at least one non-scattered, and forces $C_p(X)$2 to contain $C_p(X)$3.

Kania and Kąkol now eliminate all hypotheses on $C_p(X)$4, and simultaneously weaken the conclusion required of the map.

Statement of the main results

Theorem A. Let $C_p(X)$5 be a compact Hausdorff space and $C_p(X)$6 an infinite-dimensional real Banach space. There is no continuous bijection $C_p(X)$7 whose inverse is continuous at $C_p(X)$8.

Corollary B. If $C_p(X)$9 and $E$0 are infinite compact Hausdorff spaces, then $E$1; in particular $E$2. This resolves all three open problems above in the negative.

Two sharpness observations frame the theorem. First, bijectivity cannot be relaxed to surjectivity: Krupski and Marciszewski constructed continuous surjections from $E$3 onto $E$4 with norm topology when $E$5 is a convergent sequence and $E$6 is infinite compact metrisable. Second, the authors do not claim that continuity of $E$7 at a single point is dispensable under bijectivity; this remains open.

Ingredients of the proof

The argument combines three tools.

A linearly independent norm-compact arc in $E$8. Using Mazur's basic sequence theorem, one constructs $E$9 from a normalised basic sequence $E_w$0. The image $E_w$1 is a norm-compact arc on which norm and weak-star topologies agree, excludes $E_w$2, and—crucially—is linearly independent (Vandermonde determinism), so any finite-dimensional subspace meets it only finitely. This refines a general result of Banakh and Plichko on copies of compact metric spaces in complete linear metric spaces.

The finite-support method. For each pair $E_w$3 define

$E_w$4

where $E_w$5 is the Vietoris hyperspace of non-empty subsets of size at most $E_w$6 and $E_w$7. Compactness of $E_w$8 follows since $E_w$9 is closed under Vietoris neighbourhoods and $E$0 is norm-continuous on $E$1. Since $E$2 is continuous at $E$3, every $E$4 lies in some projection $E$5, so the Baire category theorem yields some $E$6 with interior in $E$7. Continuity of $E$8 at $E$9 further forces each fibre over a fixed support $C_p([0,1])$0 to lie in a finite-dimensional span of finitely many functionals, hence to be finite—a point where linear independence of $C_p([0,1])$1 is essential.

The pointwise-null cut-off lemma. Given pairwise disjoint finite sets $C_p([0,1])$2 of uniformly bounded size, one extracts a subsequence admitting $C_p([0,1])$3 with $C_p([0,1])$4 on $C_p([0,1])$5 and $C_p([0,1])$6 pointwise. The proof proceeds by induction on the uniform cardinality bound, using an $C_p([0,1])$7-accumulation point, pairwise disjoint supports, and Urysohn's lemma. Notably, the cardinality bound cannot be dropped: without it the conclusion would imply property $C_p([0,1])$8, which for compacta is equivalent to scatteredness, so the lemma fails outright for every non-scattered compact space.

The contradiction

The Baire argument selects minimal $C_p([0,1])$9 with $E_w$0, producing a relatively open $E_w$1 and distinct $E_w$2 in norm within $E_w$3, each paired with a $E_w$4-element support $E_w$5 (each fixed support occurs finitely often). A $E_w$6-system decomposition (the countable $E_w$7-uniform case of Erdős–Rado) reorganises the $E_w$8 into a common root $E_w$9 and pairwise disjoint petals $E$0. Minimality of $E$1 supplies $E$2 with $E$3 for large $E$4. Applying the cut-off lemma to the petals gives $E$5 on $E$6 tending pointwise to $E$7, so $E$8 pointwise yet $E$9, whence $C$0. But $C$1 is weakly null, hence norm bounded, and

$C$2

forcing $C$3—contradicting the upper bound. This contradiction establishes Theorem A directly.

Limitations and open questions

The paper states two precise boundaries of its method. First, compactness of $C$4 enters twice: it renders each $C$5 compact (so the $C$6 are closed, as required by the Baire argument), and it underpins the cut-off lemma through regularity, normality, and existence of accumulation points. Neither mechanism extends to arbitrary Tychonoff spaces, so whether $C$7 can occur for an infinite Tychonoff $C$8—Problem 3.10 of Kąkol–Leiderman–Michalak beyond the compact case—remains open. Second, whether bijectivity plus continuity alone (without continuity of $C$9 anywhere) suffices to obstruct such maps is not resolved; only the failure of the surjective version is known. Third, the paper does not determine which Tychonoff spaces $K$0 admit $K$1 at all; the necessary conditions of Kąkol–Leiderman–Michalak ($K$2 a countable union of compacta with a non-scattered member, $K$3 containing $K$4) remain the state of the art.

Conclusion

The paper proves that no continuous bijection between $K$5 and any weak Banach space can have a continuous inverse even at a single point, thereby answering Krupski's problem, its two-space variant, and the $K$6 question negatively and without auxiliary hypotheses. Technically, the result demonstrates how the classical finite-support method gains decisive leverage from a norm-compact linearly independent set of functionals combined with a Baire category argument on the dual arc. The remaining questions concern the Tychonoff case and the exact role of inverse continuity, both left explicitly unresolved.