## Return to Answer

2 added 39 characters in body; [made Community Wiki]

EDIT: According to François's comment below, this answer only works for finite fields instead of fields of positive characteristic, as I believe originally intended. I would like to leave it for a while in community wiki and see if someone else's can use these ideas to give a further step. END EDIT

Here is another proof that the ultrafilter lemma theorem is also enough to deduce statement (D) for vector spaces over finite fieldsof positive characteristic, which generalizes the answer given by François. The idea is to use partial functionals defined on finite-dimensional subspaces and with values in $\mathbb{F}_p$ (if the field has characteristic $p$, then the vector space can be considered as an $\mathbb{F}_p$-vector space by restriction) and use a consistency principle to deduce the existence of a functional defined in the whole space. This can be done by using the following theorem, which is equivalent to the prime ideal theorem (and hence to the ultrafilter lemma):

THEOREM: Suppose for each finite $W \subset I$ there is a nonempty set $H_W$ of partial functions on I whose domains include $W$ and such that $W_1 \subseteq W_2$ implies $H_{W_2} \subseteq H_{W_1}$. Suppose also that, for each $v \in I$, {$h(v): h \in H_{\emptyset}$} is a finite set. Then there exists a function $g$, with domain $I$, such that for any finite $W$ there exists $h \in H_W$ with $g|_W \subseteq h$.

This is theorem 1 in this paper by Cowen, where he proves the equivalence with the prime ideal theorem (a simple proof using compactness for propositional logic is given by the end of the paper, and is close to what François had in mind). This is essentially also the "Consistency principle" as appearing in Jech's "The axiom of choice", pp. 17, since although Jech's formulation uses only two-valued functions, the proof he gives there, through the ultrafilter lemma, actually works when the functions are $n$-valued.

Now, for an infinite-dimensional vector space $V$ of characteristic $p$, over a finite field, fix a nonzero $v_0 \in V$ and define the sets $H_W$ for $W \subset V$ as follows: if $W \subseteq U$ for a finite $U$, consider the set $S_{U}$ of all functionals defined on the (finite-dimensional) subspace generated by $U$, with values in $\mathbb{F}_p$, U$and such that$v_0 \in U \implies f(v_0)=1$. Then$H_W$is the union of all$S_U$for finite$U \supseteq W$. By the previous theorem, we have a function$f: V \to \mathbb{F}_p$, mathbb{F}$, and the restriction property shows that $f$ is linear and $f(v_0)=1$.

1

I believe that the ultrafilter lemma is also enough to deduce statement (D) for vector spaces over fields of positive characteristic, which generalizes the answer given by François. The idea is to use partial functionals defined on finite-dimensional subspaces and with values in $\mathbb{F}_p$ (if the field has characteristic $p$, then the vector space can be considered as an $\mathbb{F}_p$-vector space by restriction) and use a consistency principle to deduce the existence of a functional defined in the whole space. This can be done by using the following theorem, which is equivalent to the prime ideal theorem (and hence to the ultrafilter lemma):

THEOREM: Suppose for each finite $W \subset I$ there is a nonempty set $H_W$ of partial functions on I whose domains include $W$ and such that $W_1 \subseteq W_2$ implies $H_{W_2} \subseteq H_{W_1}$. Suppose also that, for each $v \in I$, {$h(v): h \in H_{\emptyset}$} is a finite set. Then there exists a function $g$, with domain $I$, such that for any finite $W$ there exists $h \in H_W$ with $g|_W \subseteq h$.

This is theorem 1 in this paper by Cowen, where he proves the equivalence with the prime ideal theorem (a simple proof using compactness for propositional logic is given by the end of the paper, and is close to what François had in mind). This is essentially also the "Consistency principle" as appearing in Jech's "The axiom of choice", pp. 17, since although Jech's formulation uses only two-valued functions, the proof he gives there, through the ultrafilter lemma, actually works when the functions are $n$-valued.

Now, for an infinite-dimensional vector space $V$ of characteristic $p$, fix a nonzero $v_0 \in V$ and define the sets $H_W$ for $W \subset V$ as follows: if $W \subseteq U$ for a finite $U$, consider the set $S_{U}$ of all functionals defined on the (finite-dimensional) subspace generated by $U$, with values in $\mathbb{F}_p$, and such that $v_0 \in U \implies f(v_0)=1$. Then $H_W$ is the union of all $S_U$ for finite $U \supseteq W$. By the previous theorem, we have a function $f: V \to \mathbb{F}_p$, and the restriction property shows that $f$ is linear and $f(v_0)=1$.
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