关于无穷矩阵乘法的结合性
On the Associativity of Infinite Matrix Multiplication
摘要：
无限矩阵乘积的自然定义模仿了有限矩阵乘法的通常公式，但需要注意（在没有任何收敛意义的情况下），第一个因子的每一行的支集与第二个因子的每一列的支集的交集必须是有限的。乘法因此没有完全定义，但限于无限矩阵上的特定关系。为了使三个无限矩阵$A$、$B$和$C$的乘积以关联方式表现，中间因子$B$必须以三种方式连接$A$和$C$：（i）$AB$和$BC$必须都定义;（ii）$A(BC)$和$(AB)C$必须都定义;最后，（iii）$A(BC)$必须等于$(AB)C$。在这篇文章中，这些条件进行了研究，其特征在于在各种方式通过类似于那些正式的微积分的可求和的概念。
英文摘要：
A natural definition of the product of infinite matrices mimics the usual formulation of multiplication of finite matrices with the caveat (in the absence of any sense of convergence) that the intersection of the support of each row of the first factor with the support of each column of the second factor must be finite. Multiplication is hence not completely defined, but restricted to a specific relation on infinite matrices. In order for the product of three infinite matrices $A$, $B$, and $C$ to behave in an associative manner, the middle factor, $B$, must link $A$ and $C$ in three ways: (i) $AB$ and $BC$ must both be defined; (ii) $A(BC)$ and $(AB)C$ must both be defined; and, finally, (iii) $A(BC)$ must equal $(AB)C$. In this article, these conditions are studied and are characterized in various ways by means of summability notions akin to those of formal calculus.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
