A Banach space of (Hamel) dimension κ\kappa exists if and only if κℵ0=κ\kappa^{\aleph_0}=\kappa

A Banach space of (Hamel) dimension κ exists if and only if κ0=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.

Answer

No, this is not true.

If D is a Dedekind finite set with a Dedekind finite power set, then 1(D) is a Banach space which has a Hamel basis which is also a Schauder basis, and every linear operator from 1(D) to a normed space is continuous.

But if D is Dedekind finite, then |D|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.

Attribution
Source : Link , Question Author : Sushil , Answer Author : Asaf Karagila

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