一类泛函族的共鸣定理

Resonance theorem of a class of functionals

设Ｐ（Ｘ）是集合Ｘ的非空子集全体按集合的包含关系作成的半序集，Ａ：Ｐ（Ｘ）→Ｐ（Ｘ）是增算子．Ｘ上的实泛函族｛Ｔλ｜λ∈Ａ｝称为是Ａ型的，如果对任伺满足的子集Ｅ（Ｘ）都有得到了Ａ型泛函族的共鸣定理定理１拓扑空间Ｘ的每个点点上方有界的Ａ型泛函族都在Ｘ的某相应的非空开集上一致上方有界的充要条件是：Ｘ的任何满足条件的非空子集列都至少有一个元的内部非空．

Let P(X)be the Partially ordered space of all nonempty subests of set X under the containing relation of sets. A real functiods{Tλ|λ∈A}is taled to be of A-type if sup{Tλ(x)|λ∈A,x∈A(E)}<+∞for every subest E whixch satisfies the conditionsup{Tλ(x)|∈A, x∈E} <+∞The resonance theorem of the functionals of A-atype is obtained.Theorem 1 Every pointwise upper bounded functionals of A-type on X is uniformly upper bounded in some nonempty open set,if and only if every nonempty subest sequence {Xn} which satisfies the conditionhas at least one element with nonempty interior