试验函数空间中的非标准分析

The Nonstandard Analysis in Test Function Spaces

用非标准分析方法来研究试验函数空间Ｅ（Ｒｎ）与Ｄ（Ｒｎ）上的拓扑，对其中的单子进行了刻划，得到了这两个空间中有界性，紧性等拓扑性质的非标准特征，并十分简洁地证明了几个标准结果．最后，用非标准方法得到了Ｅ（Ｒｎ）在Ｄ′（Ｒｎ）中稠密，Ｄ（Ｒｎ）在Ｅ′（Ｒｎ）中稠密的结论．

Abstract In this paper, the topology in test function spaces E(R n) and D(R n) is studied with nonstandard analysis methods. By the description of monads, we obtain the nonstandard characterizations of boundedness and compactness in the two spaces. Finally, as applications of our conclusions, we give simple proofs of a few of classical results, especially we succinctly prove that E(R n) and D(R n) are dense respectively in D ′(R n) and E ′(R n).