AI 中文总结
本文针对具有无条件基的复巴拿赫空间，证明特定排序的单项式构成其带紧开拓扑的整全纯函数空间的Schauder基，并将结果应用于多类巴拿赫序列格与向量值E-和空间。
AI 中文摘要
设X为具有无条件Schauder基的复巴拿赫空间，本文证明与该基关联的单项式，在每个齐次次数上赋予平方序、全局赋予任意相容序后，构成空间(ℋ(X),τ₀)的Schauder基，其中(ℋ(X),τ₀)是X上带紧开拓扑的整全纯函数空间。证明依赖两个核心思路：其一，坐标尾条件给出X的紧实心子集的基本系；其二，坐标上的傅里叶投影给出关于此类紧集K上每个单项式的上确界半范数的基常数的一致估计c_{K,n}≤n+1。本文将结果应用于具有稠密c₀₀的巴拿赫序列格，包括经典空间、洛伦茨空间、Orlicz-heart空间和Schreier型序列空间，以及一般的向量值E-和，特别覆盖有限维ℓₚ空间的混合c₀-和与ℓᵣ-和，以及洛伦茨预对偶d₊(w,1)。
英文摘要
Let $X$ be a complex Banach space with an unconditional Schauder basis. We prove that the monomials associated with this basis, endowed in each homogeneous degree with the square order and globally with any compatible ordering, form a Schauder basis for the space $(\calH(X),τ_0)$ of entire holomorphic functions on $X$ with the compact-open topology. The proof relies on two main ideas. First, coordinate-tail conditions yield a fundamental system of compact solid subsets of $X$. Second, Fourier projections in the coordinates give the uniform estimate $c_{K,n}\leq n+1$ for the basis constant of the degree-$n$ monomials with respect to the supremum seminorm on each such compact set $K$. We derive applications to Banach sequence lattices with dense $c_{00}$, including classical, Lorentz, Orlicz-heart and Schreier-type sequence spaces, and to general vector-valued $E$-sums. In particular, the result covers mixed $c_0$- and $\ell_r$-sums of finite-dimensional $\ell_p$ spaces and the Lorentz predual $d_*(w,1)$.
