Talk:Infinite-Dimensional Banach Space has Uncountable Dimension

Maybe "that is, $\dim X > \aleph_0$" would do here? Caliburn (talk) 15:42, 14 June 2023 (UTC)

My erring I think is because $X$ uncountable implies $\card X > \aleph_0$ is a theorem iirc? I worry saying "$\dim X$ is the cardinality of some uncountable set" might seem strange to spell out. Caliburn (talk) 15:44, 14 June 2023 (UTC)

It is Definition 2 of Dimension of Vector Space. --Usagiop (talk) 22:58, 14 June 2023 (UTC)

My beef is the fact that the theorem shows that "an arbitrary Hamel basis of an infi dim ban space is uncountable" whereas the title says that "an infi dim ban space has uncountable dimension".

The fact that one is equivalent to the other is not the point. The point is that either:

a) the theorem should say "an infi dim ban space has uncountable dimension" and change the proof to raise the Hamel basis in the body of the proof, then say at the end of the proof "We have shown the Hamel basis to be uncountable. Hence by (whatever theorem) that means $X$ is infinite-dimensional."

or:

b) the title of the theorem should be "Infinite-Dimensional Banach Space has Uncountable Hamel Basis".

--prime mover (talk) 06:04, 15 June 2023 (UTC)

Or, c) the theorem says "$\dim X > \aleph_0$". That is, each Hammel basis is uncountable." --Usagiop (talk) 20:33, 15 June 2023 (UTC)

Inadequate. It needs to state that because a Hamel base (note spelling) is uncountable, it follows that etc. etc. --prime mover (talk) 20:59, 15 June 2023 (UTC)