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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| Soggetti: | |
| Accesso online: | https://arxiv.org/abs/2512.01740 |
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Sommario:
- In [8] probabilistic methods, in particular a variant of the Weak Law of Large Numbers related to the Bernoulli distribution, have been used to show that for every infinite compact spaces K and L there exists a sequence $(μ_n)$ of normalized signed measures on $K\times L$ with finite supports which converges to $0$ with respect to the weak topology of the dual Banach space $C(K\times L).$ In this paper, we return to this construction, limiting ourselves only to elementary combinatorial calculus. The main efects of this construction are additional information about the measures $μ_n$, this is particularly clearly seen (among the others) in the resulting inequalities $$\frac{1}{2\sqrtπ}\frac{1}{\sqrt{n}} <\sup_{A\times B\subset X\times Y} |μ_n(A\times B)|<\frac{2}{\sqrtπ}\frac{1}{\sqrt{n}},$$ $n\in\mathbb{N}$, with $μ_n(f) \to_n 0$ for every $f\in C(X \times Y);$ where X and Y are arbitrary Tychonoff spaces containing infinite compact subsets, respectively. As an application we explicitly describe for Banach spaces $C(X\times Y)$ some complemented subspaces isomorphic to $c_0$. This result generalizes the classical theorem of Cembranos and Freniche, which states that for every infinite compact spaces K and L, the Banach space $C(K\times L)$ contains a complemented copy of the Banach space $c_0.$