We discuss the continuity of the composition on several spaces of holomorphic mappings on open subsets of a complex Banach space. On the Fréchet space of entire mappings that are bounded on bounded sets, the composition turns out to be even holomorphic. In such a space, we consider linear subspaces closed under left and right composition. We discuss the relationship of such subspaces with ideals of operators and give several examples of them. We also provide natural examples of spaces of holomorphic mappings where the composition is not continuous.

中文翻译：

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全纯映射空间的合成运算。

我们讨论了复杂Banach空间的开放子集上全同构映射的几个空间上的组合的连续性。在以有界集合为界的整个映射的Fréchet空间上，合成结果甚至是全纯的。在这样的空间中，我们认为线性子空间在左右构图下是封闭的。我们讨论了这类子空间与算子理想的关系，并给出了几个例子。我们还提供了组成不连续的全纯映射空间的自然示例。