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Let $B$ be a Banach space where the dimension of the underlying vector space is countable. using the Baire category Theorem, prove that the dimension of the underlying vector space is, in fact, finite.

Can you give me how I should approach this problem? Thanks

I have checked the answer of "Let X be an infinite dimensional Banach space. Prove that every Hamel basis of X is uncountable." However, It seems they prove that Hamel basis is uncountable, whereas My problem is proving the dimension of the underlying vector space is finite.

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  • $\begingroup$ From the paragraph you added is seems that it is not clear to you how the other question relates to your question. Well, proving that $X$ is not finite-dimensional implies that Hamel basis is not countable is the same as proving that if Hamel basis is countable, then $X$ is finitely-dimensional. This is called contrapositive or contraposition. $\endgroup$ – Martin Sleziak Apr 1 '16 at 7:39