On Some Basic Theorems of Continuous Module Homomorphisms between Random Normed Modules
On Some Basic Theorems of Continuous Module Homomorphisms between Random Normed Modules
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DOI：
10.1155/2013/989102
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发表时间：
2013-10
影响因子：
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通讯作者：
G. Tiexin
中科院分区：
文献类型：
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作者：
G. Tiexin
首先通过同时考虑随机赋范模的-拓扑和局部凸拓扑两种拓扑，证明了随机赋范模间模同态的共振定理、闭图定理、逆算子定理和开映射定理。然后，为了进一步发展完全随机内积模上的模同态理论，我们给出了两种拓扑下完全随机内积模上已知的正交分解定理和Riesz表示定理的一个可读性较好的证明。最后，为了将随机赋范模之间的模同态与普通赋范空间之间的线性算子连接起来，我们给出了随机共轭空间与经典共轭空间连接的已知结果的一个较好可读性的证明，即，其中和是一对具有随机赋范模的Hölder共轭数，对应的随机共轭空间为（p. 1）。，)由（resp.）派生的空间。的普通共轭空间
We first prove the resonance theorem, closed graph theorem, inverse operator theorem, and open mapping theorem for module homomorphisms between random normed modules by simultaneously considering the two kinds of topologies—the -topology and the locally -convex topology for random normed modules. Then, for the future development of the theory of module homomorphisms on complete random inner product modules, we give a proof with better readability of the known orthogonal decomposition theorem and Riesz representation theorem in complete random inner product modules under two kinds of topologies. Finally, to connect module homomorphism between random normed modules with linear operators between ordinary normed spaces, we give a proof with better readability of the known result connecting random conjugate spaces with classical conjugate spaces, namely, , where and are a pair of Hölder conjugate numbers with a random normed module, the random conjugate space of the corresponding (resp., ) space derived from (resp., ), and the ordinary conjugate space of