Two questions about vector spaces absent AC. - MathOverflow

Two questions about vector spaces absent AC.

2011-11-14T16:59:24Z

My questions are motivated by

which asks, in the absence of AC, whether a subspace of a vector space with a basis must have a basis.

Does every real vector space embed isomorphically into a vector space with a basis?

If $V$ is a vector subspace of a vector space with a basis, then clearly the linear functionals on $V$ separate points. If $V$ is a vector space s.t. the the linear functionals on $V$ do not separate points, then by modding out the intersection of the kernels of all linear functionals you get a vector space that has no non zero linear functional.

Is there a non zero real vector space on which there is no non zero linear functional?

There are models of ZF in which every linear functional on every Banach space is continuous. In ZFC there are complete linear metric spaces on which every non zero linear functional is discontinuous. So I assume that (2) has a negative answer, which would imply that (1) also has a negative answer.

Answer by François G. Dorais for Two questions about vector spaces absent AC.

2011-11-14T18:09:05Z

Hans Läuchli [Auswahlaxiom in der Algebra, Comment. Math. Helv. 37, MR0143705, DOI:10.5169/seals-28602] has constructed a model of ZFA wherein there is vector space (over any given field, see comments) which is not finite dimensional but all of its proper subspaces are finite dimensional. In particular, there cannot be any nontrivial linear functionals since the kernel of such a linear functional would have codimension 1.

This can be transferred to ZF via the Jech-Sochor Embedding Theorem.

Answer by Clinton Conley for Two questions about vector spaces absent AC.

2011-11-14T18:28:19Z

To complement François' answer, here's a fairly explicit example of a real vector space admitting no nonzero linear functional in a model of ZFDC + all sets of reals have the Baire property (which is equiconsistent with ZF).

The space is $\mathbb{R}^\mathbb{N} / E_1$, where $E_1$ is the equivalence relation of eventual agreement of real sequences. This can be given a real vector space structure in the obvious way. Now, if $f: \mathbb{R}^\mathbb{N} / E_1 \to \mathbb{R}$ is a linear function, it descends to a linear function $g: \mathbb{R}^\mathbb{N} \to \mathbb{R}$ by $g(x) = f([x]_{E_1})$. Since $g$ is Baire measurable and $E_1$-invariant, by generic ergodicity of $E_1$ it is constant on a comeager set. This constant has to be $0$. But then $g$ is a Polish group homomorphism and is automatically continuous, and thus equals $0$ everywhere.