A Banach space of (Hamel) dimension $\kappa$ exists if and only if $\kappa^{\aleph_0}=\kappa$

Question

A Banach space of (Hamel) dimension $\kappa$ exists if and only if $\kappa^{\aleph_0}=\kappa$.

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 $B^\Bbb N$ equinumerous with $B$?

Note: $B^\Bbb N$ is not empty as $B$ is specified.

Answer 1

No, this is not true.

If $D$ is a Dedekind finite set with a Dedekind finite power set, then $\ell_1(D)$ is a Banach space which has a Hamel basis which is also a Schauder basis, and every linear operator from $\ell_1(D)$ to a normed space is continuous.

But if $D$ is Dedekind finite, then $|D|^{\aleph_0}>|D|$. So it suffices to assume that an infinite Dedekind finite set like that exists. Which is of course consistent with the failure of choice.

See also:

Brunner, Norbert "Garnir's dream spaces with Hamel bases." Arch. Math. Logik Grundlag. 26 (1987), no. 3-4, 123–126.