>>773
>>>769
>反論は？

・さすがプロだね。鋭いツッコミですね
・半分自己解決したので、下記を貼ってきますね
（十分読み込んでいないのだが　；ｐ）
・要するに
　”BCT1 is used to prove that a Banach space cannot have countably infinite dimension.”
　”Banach's theorems
　Therefore, a Banach space cannot be the union of countably many closed subspaces, unless it is already equal to one of them; a Banach space with a countable Hamel basis is finite-dimensional.”
　”math.stackexchange：Let X be an infinite dimensional Banach space. Prove that every Hamel basis of X is uncountable.”

（参考）
https://en.wikipedia.org/wiki/Baire_category_theorem
Baire category theorem
BCT is used to prove Hartogs's theorem, a fundamental result in the theory of several complex variables.
BCT1 is used to prove that a Banach space cannot have countably infinite dimension.

Relation to the axiom of choice
The proof of BCT1 for arbitrary complete metric spaces requires some form of the axiom of choice; and in fact BCT1 is equivalent over ZF to the axiom of dependent choice, a weak form of the axiom of choice.[10]

A restricted form of the Baire category theorem, in which the complete metric space is also assumed to be separable, is provable in ZF with no additional choice principles.[11] This restricted form applies in particular to the real line, the Baire space
{\displaystyle \omega ^{\omega },} the Cantor space
{\displaystyle 2^{\omega },} and a separable Hilbert space such as the
{\displaystyle L^{p}}-space
{\displaystyle L^{2}(\mathbb {R} ^{n})}.

https://en.wikipedia.org/wiki/Banach_space
Banach space

Banach's theorems
Therefore, a Banach space cannot be the union of countably many closed subspaces, unless it is already equal to one of them; a Banach space with a countable Hamel basis is finite-dimensional.

検索 a Banach space with a countable Hamel basis is finite-dimensional.
https://math.stackexchange.com/questions/217516/let-x-be-an-infinite-dimensional-banach-space-prove-that-every-hamel-basis-of
Let X be an infinite dimensional Banach space. Prove that every Hamel basis of X is uncountable.
asked Oct 20, 2012 mintu