半范数的泛函表示及应用

FUNCTIONAL REPRESENTATION OF SEMINORMS AND ITS APPLICATIONS

给出了局部凸空间上连续半范数，有界半范数和下半连续半范数等的泛函表示，应用这些表示定理，我们得到了Ｂａｎａｃｈ－Ｍａｃｋｅｙ空间的一个全局特征和囿空间的对偶特征，最后还给出了局部凸空间理论中一些重要定理的简化证明。设Ｘ是Ｈａｕｓｄｏｒｆｆ局部凸空间，Ｘ′为Ｘ上的连续线性泛函全体，Ｘ￣ｂ是Ｘ上的有界线性泛函全体，则有定理１（３）ｐ：Ｘ→Ｒ是连续（下半连续）半范数当且仅当存在Ｘ′的等度连续（σ（Ｘ′，Ｘ）有界）子集Ｂ使得对任何ｘ∈Ｘ都有定理４Ｘ是Ｂａｎａｃｈ－Ｍａｃｋｅｙ空间当且仅当Ｘ上每个下半连续半范数都是有界的。定理５Ｘ是囿空间当且仅当Ｘ￣ｂ中的β（Ｘ￣ｂ，Ｘ）有界集都是Ｘ′中的等度连续集。

The functional representation of some seminorms such as the continuous seminorms, the beundedseminorms and the lower semicontinuous seminorms are given out.Applying these representation theo-rems,we obtain a global charactoristics of Banach-Mackey spaces and the dual charactoristics ofbornologic spaces, and finally put forward simple proofs for some important theorems in the theory oflocally convex spaces. Let X be a Hausdorff locally conwex space, X′the set of all continuous linearfunctions,X￣b the set of all bounded linear functions, then one has Theorem 1(3)p: X→R is a continuous(lower semicontinuous)seminorm if and ouly if there ex-ists an equicontinuous(a σ(X′,X)-bounded)subset B of X′such thatfor all x∈X.Theorem 4 X is a Banach-Mackey space if and only if every lower semicontinuous seminorm onX is bounded.Theorem 5 X is a bernologic space if and only if every β(X￣b, X)-bounded subset of X￣b is e-quicontinuous on X.