On isomorphisms with small bound
On isomorphisms with small bound
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关于小界同构
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发表时间：
1967
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通讯作者：
M. Cambern
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作者：
M. Cambern
如果X和Y是局部紧的Hausdorff空间，并且如果我们分别用Co（X）和CO（Y）表示在X和Y上无穷远处消失的连续复值函数的Banach空间，那么，根据Banach Stone定理，函数空间Co（X）和Co（Y）之间的等距的存在性意味着X和Y是同胚的。在[1 ]中证明了，如果我们另外假定X和Y满足第一可数性公理，那么这个定理可以通过考虑具有小范数的同构而不是等距来推广。本文的目的是证明文[1]的结论在没有第一可数性假设的情况下仍然有效。
If X and Y are locally compact Hausdorff spaces, and if we denote by Co(X) and CO( Y) the Banach spaces of continuous, complexvalued functions vanishing at infinity on X and Y respectively, then, according to the Banach-Stone Theorem, the existence of an isometry between the function spaces Co(X) and Co(Y) implies that X and Y are homeomorphic. In [1 ] it was shown that if we assume, in addition, that X and Y satisfy the first axiom of countability, then this theorem can be generalized by considering isomorphisms with small norm instead of isometries. The object of this paper is to show that the conclusions of [1] remain valid without the assumption of first countability.