I*S 发帖数: 203 | 1 1.5.1. Theorem. If X is a normed space, there is a Banach space Y that
contains X as an everywhere-dense subspace (and such that the norm on X is
the
restriction of the norm on Y). If Y1 is another Banach space with these
properties, the identity mapping on X extends to an isometric isomorphism
from
Y onto Y1.
证明中说X的完备化结果为X~,并且X is an everywhere-dense subset of X~(还不清
楚啥
叫everywhere-dense...);但是后面又说
We shall show that X~ can be made into a Banach space,
~~~~~~~~~~~~~ | H****h 发帖数: 1037 | 2 everywhere-dense 就是 dense 的意思,这里指X的闭包是X~。
经过完备化以后,最开始只知道X~是一个完备的度量空间。
需要加入线性结构和定义范数,才能使之成为完备的赋范空间。
【在 I*S 的大作中提到】![](/moin_static193/solenoid/img/up.png) : 1.5.1. Theorem. If X is a normed space, there is a Banach space Y that : contains X as an everywhere-dense subspace (and such that the norm on X is : the : restriction of the norm on Y). If Y1 is another Banach space with these : properties, the identity mapping on X extends to an isometric isomorphism : from : Y onto Y1. : 证明中说X的完备化结果为X~,并且X is an everywhere-dense subset of X~(还不清 : 楚啥 : 叫everywhere-dense...);但是后面又说
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