ϵ-Weak Cauchy sequences and a quantitative version of Rosenthal's -theorem

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• 作者：I. Gasparis ^{ioagaspa@math.ntua.gr" class="auth_mail" title="E-mail the corresponding author}
• 关键词：The ℓ_{1ℓ1-theorem} ; Weak Cauchy sequence ; Ramsey theorem
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2016
• 出版时间：15 February 2016
• 年：2016
• 卷：434
• 期：2
• 页码：1160-1165
• 全文大小：286 K

文摘

A bounded sequence 5447853ae7a" title="Click to view the MathML source">(x_n)$(x_n)$ in a Banach space is called ϵ  -weak Cauchy, for some ϵ>0$ϵ>0$, if for all x^⁎∈B_X^⁎$x^{⁎}\in B_{X^{⁎}}$ there exists some n₀∈N$n_0\in \mathbb{N}$ such that e177926221a5e91416a691ac4" title="Click to view the MathML source">|x^⁎(x_n)−x^⁎(x_m)|<ϵ$|x^{⁎}(x_n)-x^{⁎}(x_m)|<ϵ$ for all n≥n₀$n\ge n_0$ and m≥n₀$m\ge n_0$. It is shown that given ϵ>0$ϵ>0$ and a bounded sequence 5447853ae7a" title="Click to view the MathML source">(x_n)$(x_n)$ in a Banach space then either 5447853ae7a" title="Click to view the MathML source">(x_n)$(x_n)$ admits an ϵ  -weak Cauchy subsequence or, for all δ>0$\delta >0$, there exists a subsequence (x_mn)$(x_{m_n})$ with the following property. If I   is a finite subset of N$\mathbb{N}$ and ϕ:I→N∖I$\varphi :I\to \mathbb{N}\setminus I$ is any map then for every sequence of complex scalars (λ_n)_n∈I${({\lambda }_n)}_{n\in I}$. This provides an alternative proof for Rosenthal's ℓ₁${\ell }_1$-theorem and strengthens its quantitative version due to Behrends. As a corollary we obtain that for any uniformly bounded sequence (f_n)$(f_n)$ of complex-valued functions, continuous on the compact Hausdorff space K   and satisfying lim⁡sup_n,m→∞|f_n(t)−f_m(t)|≤ϵ$\lim {\sup }_{n,m\to \infty }|f_n(t)-f_m(t)|\le ϵ$, for some ϵ>0$ϵ>0$ and all t∈K$t\in K$, there exists a subsequence (f_jn)$(f_{j_n})$ satisfying 4e138a3677b13523e33b6f1" title="Click to view the MathML source">lim⁡sup_n,m→∞|∫_K(f_jn−f_jm)dμ|≤2ϵ$\lim {\sup }_{n,m\to \infty }|{\int }_K(f_{j_n}-f_{j_m})d\mu |\le 2ϵ$, for every Radon measure μ on K.