论向量拟尺度空间的完备性

On the Completeness of Vector Pseudo-metric Space

δ是向量空间〈X,K〉上的平衡不变拟尺度,它定义向量拓扑τ。把δ延拓成δ*,得〈X,δ〉的完备化空间〈X1,δ*〉;δ*在〈X1,K〉上定义向量拓扑τ*,〈X1,K,δ*〉又成为〈X,K,τ〉的一致完备化空间。把τ延拓成τ*,得〈X,K,τ〉的一致完备化空间〈X*,K,τ*〉;同上法把δ延拓成X*上的δ*,δ*定义的拓扑与τ*等价,〈X*,δ*〉又成为〈X,δ〉的完备化空间。

Suppose that 〈X,K,τ〉 is a vector topological space.The topology τ is induced onto X by a translation invariant and balanced pseudo-metric δ defined on X.Then δ may be extended to have a broader pseudo-metric space 〈X1,δ＊〉 which is a completion of 〈X,δ〉 and δ＊ induces a vector topology τ＊ onto X1 to make 〈X,K,τ＊〉 be a completion of 〈X,K,τ〉.The 〈X,K,τ〉,δ ave the same as foregoing.The τ may be extended to have a broader vector topological space 〈X＊,K,τ＊〉 which is a completion of 〈X,K,τ〉.The δ extended to X＊ as foregoing onto X＊ to induce a vector topology equivalent to τ＊ and 〈X＊,δ＊〉 is a completion of 〈X,δ〉.