In this paper, the compact approximation property on Fréchet spaces is characterized in terms of holomorphic mappings. We show that a Fréchet space E has the compact approximation property if and only if every holomorphic mapping on a balanced open subset \(U\subset E\) with values in a Fréchet space can be approximated uniformly on compact subsets of U by compact holomorphic mappings. This extends the well-known linear characterization to the holomorphic setting. We also give characterizations of the compact approximation property in terms of bounded holomorphic mappings on Banach spaces.

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Fréchet空间上全纯映象空间的紧逼近性质

在本文中，Fréchet空间的紧逼近性质用全纯映射表示。我们表明，Fréchet可空间ê具有紧凑的近似性当且仅当在平衡开子集每一个全纯映射\（U \子集Ë\）在一个Fréchet可空间值可以均匀地对紧子集近似ü通过紧凑的全纯映射。这将众所周知的线性表征扩展到全纯设置。我们还根据Banach空间上的有界同胚映射给出了紧逼近性质的刻画。