论文标题
Ramsey的特性和Banach空间中标准化序列的块振荡稳定性
Ramsey Property and Block Oscillation Stability on Normalized Sequences in Banach Spaces
论文作者
论文摘要
Ramsey定理在Banach空间理论中的一个众所周知的应用是一个事实证明，每个归一化的基本序列都有一个产生扩散模型的子序列（Brunel-Sucheston定理）。基于此应用程序，作为中间步骤，我们可以讨论$（k，\ varepsilon） - $振荡稳定序列的概念，将在本文中更广泛地描述和分析。实际上，我们介绍了概念$（（\ Mathcal {b} _i）_ {i = 1}^k，\ varepsilon） - $ block振荡稳定序列，其中$（\ nathcal {b} _i）_ {i = 1}^k $是barrier和barrier barrier and barrier and barrier barrier and barriers and barrier barrier barriers and barrier barrier barrier barrier barrier barrier barrier barrier barre blocks。特别是，我们证明了Ramsey定理等同于``对于每个有限序列$（\ Mathcal {b} _i）_ {i = 1}^k $的障碍，每一个$ \ varepsilon> 0 $ 0 $和每个正常的序列$（x_i） $（x_i）_ {i \ in m} $，即$（（\ Mathcal {b} _i \ cap \ cap \ mathcal {p}（m））_ {i = 1}^k，\ varepsilon） - $ block oscillation oscillation stable stable stable''，$ \ mathcal Infer in Infers M. $（\ MATHCAL {B} _i）_ {i \ in \ Mathbb {n}} - $ block的归一化基本序列的差异分散模型，其中$（\ Mathcal {b} _i）_ {i \ in \ Mathbb {n}} $是Barriers的序列。这些模型是扩展模型的概括，与$（（\ Mathcal {b} _i）_ {i = 1}^k，\ varepsilon） - $ block振荡稳定序列。我们表明，Brunel-Sucheston对$（\ Mathcal {b} _i）_ {i \ in \ Mathbb {n}} - $ block渐近模型感到满意，我们还证明了此结果与Ramsey Theorem相当。我们的定理和Brunel-Sucheston定理之间的差异基于通过它们从相同的归一化基本序列获得的不同模型的数量。在文章结尾的一个示例中指出了有关$（\ mathcal {b} _i）_ {i \ in \ mathbb {n}}}的_ {i \ in \ mathbb {n}} _ {i \ in \ mathbb {i \ in \ mathcal {b} _i）的观察结果和其他观察结果。
A well-known application of the Ramsey Theorem in the Banach Space Theory is the proof of the fact that every normalized basic sequence has a subsequence which generates a spreading model (the Brunel-Sucheston Theorem). Based on this application, as an intermediate step, we can talk about the notion of $(k,\varepsilon)-$oscillation stable sequence, which will be described and analyzed more generally in this article. Indeed, we introduce the notion $((\mathcal{B}_i)_{i=1}^k,\varepsilon)-$block oscillation stable sequence where $(\mathcal{B}_i)_{i=1}^k$ is a finite sequence of barriers and using what we will call blocks of barriers. In particular, we prove that the Ramsey Theorem is equivalent to the statement ``for every finite sequence $(\mathcal{B}_i)_{i=1}^k$ of barriers, every $\varepsilon>0$ and every normalized sequence $(x_i)_{i\in\mathbb{N}}$ there is a subsequence $(x_i)_{i\in M}$ that is $((\mathcal{B}_i\cap\mathcal{P}(M))_{i=1}^k,\varepsilon)-$block oscillation stable'', where $\mathcal{P}(M)$ is the power set of the infinite set M. Besides, we introduce the $(\mathcal{B}_i)_{i\in\mathbb{N}}-$block asymptotic model of a normalized basic sequence where $(\mathcal{B}_i)_{i\in\mathbb{N}}$ is a sequence of barriers. These models are a generalization of the spreading models and are related to the $((\mathcal{B}_i)_{i=1}^k,\varepsilon)-$block oscillation stable sequences. We show that the Brunel-Sucheston is satisfied for the $(\mathcal{B}_i)_{i\in\mathbb{N}}-$block asymptotic models, and we also prove that this result is equivalent to the Ramsey Theorem. The difference between our theorem and the Brunel-Sucheston Theorem is based on the number of different models that are obtained from the same normalized basic sequence through them. This and other observations about $(\mathcal{B}_i)_{i\in\mathbb{N}}-$block asymptotic models are noted in an example at the end of the article.
