Existence of vector space complement and axiom of choice
Let say we live in the category of vector spaces over $\mathbb{R}$ or $\mathbb{C}.$ Here are three sentences:
- Axiom of choice
- Every vector space has a base.
- For every vector space $V$ and its subspace $E\subset V$ there is a subspace $F\subset V$ such that $V=E\oplus F.$
I know how to prove that (1)->(2)->(3). How about the inverese? Do (2)->(1) and (3)->(2) hold?
If this is not the case, then is there some weaker version of AC which imply (3)?
No, there is no weaker choice principle implying (3). It was shown that (3) implies the axiom of choice in $\sf ZF$.
The proof is via an equivalent of the axiom of choice called "The Axiom of Multiple Choice". You can find the details in Rubin & Rubin's "Equivalents of the Axiom of Choice II" as Theorem 6.35 (pp. 119-120 and 122).
The proof is due to Bleicher from 1964
M. N. Bleicher, Some theorems on vector spaces and the axiom of choice, Fund. Math. 54 (1964), 95--107.
It is interesting to note that in a more relaxed setting where there might be atoms (non-set objects) or that the axiom of regularity fails, it is not known whether or not (3) implies the axiom of choice.