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中文摘要：
证明每个F^＊空间（即满足第一可数公理的Hausdorff拓扑向量空间）可借助于它的“标准生成伪范数族”来表征,利用标准生成伪范数族P,在F^＊空间中引入P-有界集、P-半有界集和P-无界集的概念,建立点态半有界和非无界线性算子族的共鸣定理,作为其推论,得到了Menger概率赋范空间中点态概率半有界和非概率无界线性算子族的共鸣定理,改进并推广了某些已有的结果.
英文摘要：
We prove that every F^＊ space （i.e., Hausdorff topological vector space satisfying the first countable axiom） can be characterized by meams of its ＂standard generating family of pseudo-norms＂. By using the standard generating family of pseudo-norms P, the concepts of P-bounded set, P-semi-bounded set and ：P-unbounded set in F＊ space are introduced. The resonance theorems for families of pointwise semi-bounded and pointwise non-unbounded linear operators on F＊ spaces are established. As their applications, the resonance theorems for fami- lies of pointwise probabilistically semi-bounded and pointwise probabilistically non-unbounded linear operators in Menger probabilistic normed spaces are obtained.
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