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Great answer by Asaf Karagila to my question leads me to further questions.

Let say we deal with vector spaces over $\mathbb{R}.$ Here are three sentences:

  1. For every $V$ and its subspace $E\subset V$ there is a subspace $F\subset V$ such that $V=E\oplus F.$
  2. For every $V,W,$ subspace $E\subset V$ and linear map $f:E\to W$ there is a linear map $F:V\to W$ such that $F|_E=f.$
  3. For every $V,$ subspace $E\subset V$ and linear map $f:E\to\mathbb{R}$ there is a linear map $F:V\to\mathbb{R}$ such that $F|_E=f.$

Obviously (1)->(2)->(3). How about the converse? Do (3)->(2) and (2)->(1)?

From answer to my previous question we know that AC<->(1). For me, (3) looks like unnormed version of Hahn-Banach theorem. But as we read on wiki, H-B does not imply (3).

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    I'd begin by looking into Morillon's paper mentioned in my answer here, and by reading this answer of Martin. In short, we know some things about linear functionals in ZF, but not a whole lot of things. Most likely neither implication is reversible, but I think if any of them is known, it might only be that (3) does not imply (1). But I don't know. – Asaf Karagila Mar 25 '16 at 22:57

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