infinite-dimensional拓扑向量空间中的一个极小极大定理
A minimax theorem in infinite-dimensional topological vector spaces
摘要：
本文得到了一个极小极大定理，利用它证明了如下结果：设$E$是无穷维自反实Banach空间，$T:E\to E$是非零紧线性算子，$φ:E\to {\bf R}$是下连续凸强制泛函，$I\subset {\bf R}$是紧区间，$0\in I$，$ψ:I\to {\bf R}$是下连续凸函数.然后，对于每个$r>φ(0)$，具有$$\sup_{x\in X}\inf_{λ\in I}(φ(T(x)-λx)+ψ(λ))=r+ψ(0)\ ,$$，其中$$X=\{x\in E : φ(T(x))\leq r\}\ .$$
英文摘要：
In this paper, we obtain a minimax theorem by means of which, in turn, we prove the following result: Let $E$ be an infinite-dimensional reflexive real Banach space, $T:E\to E$ a non-zero compact linear operator, $φ:E\to {\bf R}$ a lower semicontinuous, convex and coercive functional, $I\subset {\bf R}$ a compact interval, with $0\in I$, $ψ:I\to {\bf R}$ a lower semicontinuous convex function. Then, for each $r>φ(0)$, one has $$\sup_{x\in X}\inf_{λ\in I}(φ(T(x)-λx)+ψ(λ))=r+ψ(0)\ ,$$ where $$X=\{x\in E : φ(T(x))\leq r\}\ .$$
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