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】
: 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
|
|
