The ultimate goal in this article is to introduce the concepts of \(\otimes\)-amenability and weak \(\otimes\)-amenability of a Banach coalgebra \(\mathfrak {C}\) and examine the dependence of these properties with the amenability and weak amenability of the Banach algebra \(\mathfrak {C}^{*}\). For this purpose, we first study the Banach \(\mathfrak {C}\)-comodule \(\mathcal {M}\) and its dual as a Banach \(\mathfrak {C}^{*}\)-module. Then, the relationship between a bounded coderivation on \(\mathcal {M}\) and its dual, as a bounded derivation on \(\mathfrak {C}^{*}\), is discussed. Also, similar conclusions are obtained for the case that A is a finite dimensional Banach algebra and \(A^{*}\) is a Banach coalgebra. Especially we investigate modules (and derivations) which arise from a comodule ( and a coderivation ) and vice versa.

本文的最终目标是介绍Banach 余代数\(\mathfrak {C}\)的\(\otimes\) -amenability 和弱\(\otimes\) -amenability 的概念，并检查这些属性的依赖性具有 Banach 代数\(\mathfrak {C}^{*}\)的顺从性和弱顺从性。为此，我们首先研究 Banach \(\mathfrak {C}\) -comodule \(\mathcal {M}\)及其对偶作为 Banach \(\mathfrak {C}^{*}\) -module . 然后，\(\mathcal {M}\)上的有界共推导与其对偶之间的关系，作为\(\mathfrak {C}^{*}\)上的有界推导, 进行讨论。此外，对于A是有限维 Banach 代数和\(A^{*}\)是 Banach 余代数的情况，也得到了类似的结论。特别是我们研究了由协模块（和代码派生）产生的模块（和派生），反之亦然。