Intereting Posts

Are mini-Mandelbrots known to be found in any fractals other than the Mandelbrot set itself?
Find a matrix equation equivalent to $A^TPA+P=I$
Can a circle's circumference be divided into arbitrary number of equal parts using straight edge and compass only?
simplify cos 1 degree + cos 3 degree +…+cos 43 degree?
About $ \{ x \in^{\omega_1}:|\{\alpha<\omega_{1} :x(\alpha)\ne 0 \}|\le\omega \}$
degree 3 Galois extension of $\mathbb{Q}$ not radical
Baby Rudin: Chapter 1, Problem 6{d}. How to complete this proof?
Deducing formulas of analytic geometry
Proving formula for product of first n odd numbers
Maximal gaps in prime factorizations (“wheel factorization”)
Nilpotency of the Jacobson radical of an Artinian ring without Axiom of Choice
On the number of caterpillars
Soft Question: Why does the Axiom of Choice lead to the weirdest constructions?
Find a finite generating set for $Gl(n,\mathbb{Z})$
Confusion about the null (empty) set being contained in other sets

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 $B$?

- Book/Books leading up to the the axiom of choice?
- Krull-Akizuki theorem without Axiom of Choice
- Mean value theorem and the axiom of choice
- For any two sets $A,B$ , $|A|\leq|B|$ or $|B|\leq|A|$
- Which set is unwell-orderable?
- Axiom of Choice and finite sets

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

- Proving existence of a surjection $2^{\aleph_0} \to \aleph_1$ without AC
- Inclusion of $\mathbb{L}^p$ spaces, reloaded
- Was Grothendieck familiar with Stone's work on Boolean algebras?
- What's an example of a vector space that doesn't have a basis if we don't accept Choice?
- Vector, Hilbert, Banach, Sobolev spaces
- Prove a certain property of linear functionals, using the Hahn-Banach-Separation theorems
- Let $X$ be an infinite dimensional Banach space. Prove that every Hamel basis of X is uncountable.
- $\ell^1$ vs. continuous dual of $\ell^{\infty}$ in ZF+AD
- Banach-Stone Theorem
- Proof of Eberlein–Smulian Theorem for a reflexive Banach spaces

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.

- Derivative of piece-wise function given by $x\sin\frac1x$ at $x=0$
- If $f$ is integrable and bounded over $$ for all $\delta \in (0,1]$ then is $f$ integrable over all of $$ $?$
- Closure of a connected subset of $\mathbb{R}$ is connected?
- Topological space definition in terms of open-sets
- Square-root for nonnegative (and not necessarily self-adjoint) operators
- Generalization of real induction for topological spaces?
- A novelty integral for $\pi$
- About rationalizing expressions
- The first homology group $ H_1(E(K); Z) $ of a knot exterior is an infinite cyclic group which is generated by the class of the meridian.
- uniqueness of solutions of $ax=b$ and $ya=b$ in a semigroup .
- Applications of Probability Theory in pure mathematics
- How to get rid of the integral in this equation $\int\limits_{x_0}^{x}{\sqrt{1+\left(\dfrac{d}{dx}f(x)\right)^2}dx}$?
- Why is Completeness not a Topological Property?
- $\int_C \frac{\log z}{z-z_0} dz$ – Cauchy theorem with $z_0$ outside the interior of $\gamma$
- Coordinate-free proof of $\operatorname{Tr}(AB)=\operatorname{Tr}(BA)$?