5
votes
1answer
130 views

Intuition for “the existence of a basis for every vector space is equivalent to the Axiom of Choice”?

Is there a intuitive way to understand "the existence of a basis for every vector space is equivalent to the Axiom of Choice"?
1
vote
2answers
177 views

Easy visualizations of small countable ordinals

The ordinal number $\omega^2$ can be visualized as $\omega$-many copies of $\omega$. Likewise, the ordinal number $\omega^3$ can be visualized as $\omega^2$-many copies of $\omega$, arranged as ...
3
votes
1answer
62 views

Cardinality of a vector space versus the cardinality of its basis

Let $V$ an infinite dimensional vector space. How to show that the cardinality of $V$ is the same of a basis of $V$? I saw this argument here link in the main answer (MathOverFlow).
1
vote
3answers
144 views

A Vector Space is a Set - Axiom or Derivation?

I understand that structures with the properties of the real and complex numbers can be defined and derived from the axioms of ZFC set theory. But can a structure with the properties of a (possibly ...
2
votes
1answer
88 views

Hamel basis dense in the unit sphere

I know that a Hamel basis can be dense in a Banach space (it was probably posted somewhere on this forum). I would like to construct a certain counter-example and doing this, I encountered the ...
15
votes
1answer
737 views

Cardinality of a Hamel basis

What is the cardinality of a Hamel basis of $\ell_1(\mathbb{R})$? Is it deducible in ZFC that it is seemingly continuum? Does it follow from this that each Banach space of density $\leqslant ...