A Banach space of (Hamel) dimension κ exists if and only if κℵ0=κ.
How will we prove the converse implication. One sided implication for Hilbert Space is proved in question: Can you equip every vector space with a Hilbert space structure?
If we don’t assume Axiom of Choice, and we have a Banach space with (Hamel Basis B existence given). Will it be true BN equinumerous with B?
Note: BN is not empty as B is specified.
No, this is not true.
If D is a Dedekind finite set with a Dedekind finite power set, then ℓ1(D) is a Banach space which has a Hamel basis which is also a Schauder basis, and every linear operator from ℓ1(D) to a normed space is continuous.
But if D is Dedekind finite, then |D|ℵ0>|D|. So it suffices to assume that an infinite Dedekind finite set like that exists. Which is of course consistent with the failure of choice.
Brunner, Norbert “Garnir’s dream spaces with Hamel bases.”
Arch. Math. Logik Grundlag. 26 (1987), no. 3-4, 123–126.