This paper is on general methods of convergence and summability. We first present the general method of convergence described by free filters of $\mathbb{N} $ and study the space of convergence associated with the filter. We notice that $c(X)$ is always a space of convergence associated with a filter (the Frechet filter); that if X is finite dimensional, then $\ell _{\infty }(X)$ is a space of convergence associated with any free ultrafilter of $\mathbb{N} $ ; and that if X is not complete, then $\ell _{\infty }(X)$ is never the space of convergence associated with any free filter of $\mathbb{N} $ . Afterwards, we define a new general method of convergence inspired by the Banach limit convergence, that is, described through operators of norm 1 which are an extension of the limit operator. We prove that $\ell _{\infty }(X)$ is always a space of convergence through a certain class of such operators; that if X is reflexive and 1-injective, then $c(X)$ is a space of convergence through a certain class of such operators; and that if X is not complete, then $c(X)$ is never the space of convergence through any class of such operators. In the meantime, we study the geometric structure of the set $\mathcal{HB}(\lim ):= \{T\in \mathcal{B} (\ell _{\infty }(X),X): T|_{c(X)}= \lim \text{ and }\|T\|=1\}$ and prove that $\mathcal{HB}(\lim )$ is a face of $\mathsf{B} _{\mathcal{L}_{X}^{0}}$ if X has the Bade property, where $\mathcal{L}_{X}^{0}:= \{ T\in \mathcal{B} (\ell _{\infty }(X),X): c_{0}(X) \subseteq \ker (T) \} $ . Finally, we study the multipliers associated with series for the above methods of convergence.

中文翻译：

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收敛和求和的一般方法

本文介绍了收敛和可加性的一般方法。我们首先介绍由$ \ mathbb {N} $的免费过滤器描述的一般收敛方法，并研究与过滤器关联的收敛空间。我们注意到，$ c（X）$始终是与过滤器（Frechet过滤器）关联的会聚空间；如果X是有限维的，则$ \ ell _ {\ infty}（X）$是与$ \ mathbb {N} $的任何免费超滤器相关的收敛空间；并且如果X不完整，则$ \ ell _ {\ infty}（X）$永远不会是与$ \ mathbb {N} $的任何免费过滤器相关联的收敛空间。然后，我们定义了受Banach极限收敛启发的一种新的通用收敛方法，即通过范数1的算子来描述，范数1是极限算子的扩展。我们证明$ \ ell _ {\ infty}（X）$始终是通过此类算子的特定类收敛的空间；如果X是自反的和单射的，则$ c（X）$是通过特定类的此类算子的收敛空间；并且如果X不完整，则$ c（X）$永远不是通过此类运算符的任何类进行收敛的空间。同时，我们研究集合$ \ mathcal {HB}（\ lim）：= \ {T \ in \ mathcal {B}（\ ell _ {\ infty}（X），X）的几何结构：T | _ {c（X）} = \ lim \ text {和} \ | T \ | = 1 \} $并证明$ \ mathcal {HB}（\ lim）$是$ \ mathsf {B}的头像_ {\ mathcal {L} _ {X} ^ {0}} $如果X具有Bade属性，则$ \ mathcal {L} _ {X} ^ {0}：= \ {T \ in \ mathcal {B }（\ ell _ {\ infty}（X），X）：c_ {0}（X）\ subseteq \ ker（T）\} $。最后，我们针对上述收敛方法研究了与级数相关的乘数。