关于闭图像定理的一个注记

A Note on the Closed Graph Theorem

闭图像定理表明Banach空间中线性算子图像的闭性蕴含算子的连续性.相反地,一个更一般的结论是从拓扑空间到Hausdorff空间的映射的连续性蕴含其图像的闭性.本文通过举例说明了其逆命题不成立,特别强调了值域空间的Haudorff分离性的重要性.此外,利用此结论证明了Banach空间中线性算子对于弱拓扑的连续性与对于强拓扑的连续性等价.最后,通过对值域空间附加紧致性条件建立了从一个拓扑空间到一个紧致Hausdorff空间的映射的连续性与其图像闭性之间的等价刻画.

The closed graph theorem shows that the closedness of the graph of the linear operator in Banach spaces implies its continuity.On the contrary,a more general conclusion is that the continuity of the mapping from a topological space to a Hausdorff space implies the closedness of its image.This paper provides several counterexamples to illustrate the inverse proposition is not true,and especially emphasizes the importance of the Haudorff separation of the range space.In addition,this conclusion is used to prove that the continuity of linear operators in Banach space for the weak topology is equivalent to the continuity for the strong topology.Finally,an equivalent characterization between the continuity of the mapping from a topological space to a compact Hausdorff space and the closedness of its image is established by attaching a compactness condition to the range space.