Can a infinite dimensional Banach space have a dense Hamel basis

Asked Dec 07 '22 at 15:20

Active Dec 07 '22 at 15:26

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I'm looking for a link of a book or paper that has the following

Theorem:

Let $X$ be an infinite dimensional Banach space. Then there exists a Hamel basis $B$ of $X$ that is dense in $X$ ( That is $\bar{B}=X$)

I found the theorem above in a textbook of functional analysis, but it only stated the theorem without proof or any other mentions. I've been searching for it for a while but i can't find it anywhere.
If someone knows where i can find it.

Thank you!

edited Dec 07 '22 at 15:26

Thomas Andrews

177,126

asked Dec 07 '22 at 15:20

lebong66

1

Does this answer your question? Is it true that Hamel basis of $C[a,b]$ can be dense in $C[a,b]$?. the answer there has proof of that theorem – Mor A. Dec 07 '22 at 15:27
@MorA. Yes it has the theorem that i'm searching for ( Theorem 1 in the link )(I've posted a question asking for it), but there isn't any proof or mentions. I'm searching for the general case which is exactly the Theorem 1 – lebong66 Dec 07 '22 at 15:41
2

The proof is right there in the answer – Mor A. Dec 07 '22 at 15:43

Can a infinite dimensional Banach space have a dense Hamel basis

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