AI 中文总结
本文开发了B-乘子空间的通用框架，研究其可赋范性、完备性等性质，引入相关乘子空间，明确BK空间关联空间概念，推导序列空间的b-理想部分结论并证明$M(\Sigma c(X),\Sigma c(X)) = bv(\mathbb{K})$。
AI 中文摘要
我们开发了一个通用框架用于B-乘子空间；这类空间是由双线性算子$B \colon M \times V \longrightarrow W$得到的向量空间$M = M(V,W)$，其中$V$和$W$是Banach空间。我们重点关注它们的可赋范性和完备性，主要针对取值于Banach空间$X$的函数空间，特别是序列空间的情形。我们的方法依赖于基础Banach空间满足BK性质，即具有连续赋值。当$B$是标量函数的逐点乘积时，就得到经典的乘子空间，我们还特别讨论了$V$或$W$由向量函数组成的情形。我们重点关注序列空间$\Sigma \ell_{\infty}(X)$（有界部分和）、$\Sigma c(X)$（可和）和$\ell_u(X)$（无条件可和）。给定一个BK标量序列空间$V$，我们引入乘子空间$M_{\Sigma}(V,X)$，并确定其满足的条件，使得该空间能确定从$V$到$X$的有界线性算子的闭子空间。我们明确了BK空间的关联空间概念，将该构造与经典Köthe对偶联系起来。我们考虑Banach序列理想$Y$的所谓强向量化$Y(X)$和弱向量化$Y_w(X)$，弱向量化可作为乘子空间得到，并用于描述经典序列空间如$\ell_{p,w}(X)$（$1 \leq p \leq \infty$）。我们引入空间的b-理想部分，证明$\Sigma c(X)$的b-理想部分是$\ell_u(X)$，$\Sigma \ell_{\infty}(X)$的b-理想部分是$\ell_{1,w}(X)$。我们还研究了序列空间$bv(X)$（有界变差），证明$M(\Sigma c(X),\Sigma c(X)) = bv(\mathbb{K})$。
英文摘要
We develop a general framework for $B$-multiplier spaces; these are vector spaces $M = M(V,W)$ obtained from a bilinear operator $B \colon M \times V \longrightarrow W$, where $V$ and $W$ are Banach spaces. We focus on their normability and completeness, mainly in the setting of spaces consisting of functions with values in a Banach space $X$, particularly sequences. Our approach relies on the underlying Banach spaces satisfying the BK property, that is, having continuous evaluations. Classical multiplier spaces arise when $B$ is a pointwise product of scalar functions and we make the point for the case when $V$ or $W$ consists of vector functions. Special attention is given to the sequence spaces $Σ\ell_{\infty}(X)$ (bounded partial sums), $Σc(X)$ (summable), and $\ell_u(X)$ (unconditionally summable). Given a BK scalar sequence space $V$, we introduce the multiplier space $M_Σ(V,X)$ and establish conditions under which it determines a closed subspace of bounded linear operators from $V$ into $X$. The notion of associate space is precised for BK-spaces, linking this construction with classical Köthe duality. We consider what we named strong vectorialization $Y(X)$ and weak vectorialization $Y_w(X)$ of a Banach sequence ideal $Y$. The weak vectorialization is obtained as a multiplier space and employed to describe classical sequence spaces as $\ell_{p,w}(X)$, $1 \leq p \leq \infty$. We introduce the $b$-ideal part of a space and show that the $b$-ideal part of $Σc(X)$ is $\ell_u(X)$ and that of $Σ\ell_{\infty}(X)$ is $\ell_{1,w}(X)$. We also study the sequence space $bv(X)$ (bounded variation), proving that $M(Σc(X),Σc(X)) = bv(\mathbb K)$.
