桶空间的几个特征性质

Several characteristics of barrelled spaces

从Ｂａｎａｃｈ－Ｓｔｅｉｎｈａｕｓ定理、算子空间的完备性和双线性映射等方面给出了桶空间的几个特征性质．主要结果是定理１设Ｘ是Ｍａｃｋｅｙ空间，Ｙ是非零的Ｈａｕｓｄｏｒｆｆ局部凸空间．则Ｘ是根空间当且仅当Ｌｓ（Ｘ，Ｙ）中任何有界网｛Ｔａ｝的点点极限Ｔ都属于Ｌｓ（Ｘ，Ｙ）．定理２设Ｘ是Ｍａｃｋｅｙ空间，Ｙ是有界完备的非零Ｈａｕｓｄｏｒｆｆ局部凸空间．则Ｘ是桶空间当且仅当Ｌｓ（Ｘ，Ｙ）是有界完备的．定理４设Ｘ和Ｙ是非零的Ｈａｕｓｄｏｒｆｆ局部凸空间，则Ｘ是桶空间当且仅当每个点点有界的从Ｘ×Ｘ到Ｙ的各别连续双线性映射族都是等度亚连续的．

Several characteristics are give out for barrelled spaces in the Banach-Steinhaus Theorem, the completeness of operator space and bilinear mappings.The main results are Theorem 1 Let X be a Mackey sauce and Y a nonzero Hausdorff locally convex space. Then X is barrelled if and only if the pointwise limit T belongs to Ls(X, Y) for every bounded net {Ta}in Ls(X,Y).Theorem 2 Let X be a Mackey space and Y a nonzero quasi-complete Hausdorff locally convex space.Then X is barrelled if and only if Ls(X,Y)is quasi-complete.Theorem 4 Let X and Y be nonzero Hausdorff locally convex spaces.Then X is barrelled if and only if every pointwise bounded family of seperately continuous bilinear mappings is equi-subcontinuous.