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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 $B$?

- Bases of complex vector spaces and the axiom of choice
- Why can't you pick socks using coin flips?
- Zorn's Lemma And Axiom of Choice
- Why is axiom of dependent choice necesary here? (noetherian space implies quasicompact)
- Why is the axiom of choice separated from the other axioms?
- Does a nonlinear additive function on R imply a Hamel basis of R?

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

- Norms on C inducing the same topology as the sup norm
- $a\mapsto \log\left(\lVert f\lVert_{1/a}\right)$ is a convex map
- Weak limit of an $L^1$ sequence
- Non-aleph infinite cardinals
- Finding a choice function without the choice axiom
- If $V \times W$ with the product norm is complete, must $V$ and $W$ be complete?
- Compact operators, injectivity and closed range
- Kuratowski-Zorn Lemma with pre-order (quasi-order, proset) instead poset?
- Bochner: Lebesgue Obsolete?
- A Hamel basis for $l^{\,p}$?

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.

- Parseval's Identity (Integral)
- Is “A and B imply C” equivalent to “For all A such that B, C”?
- $FTC$ problem $\frac d{dx}\int_1^\sqrt xt^tdt$
- For what value of h the set is linearly dependent?
- Size of a union of two sets
- Is there a $4$-regular planar self-complementary graph with $9$ vertices and $18$ edges?
- Convergence of “alternating” harmonic series where sign is +, –, +++, —-, etc.
- When are complex numbers insufficient?
- Why is $A_5$ a simple group?
- Bijecting a countably infinite set $S$ and its cartesian product $S \times S$
- Baker-Hausdorff Lemma from Sakurai's book
- Problem understanding “and”,“or” and importance of “()” in set theory
- Showing $1+p$ is an element of order $p^{n-1}$ in $(\mathbb{Z}/p^n\mathbb{Z})^\times$
- Using Leibniz Integral Rule on infinite region
- Understanding Arzelà–Ascoli theorem