Injective Banach spaces, with morphisms as contractive linear maps, have been classically studied (and are $C(K)$ spaces with $K$ Stonian). But what about projectives?


So $P$ will be projective if given Banach spaces $E$ and $F$ and a quotient map (aka metric surjection) $\psi:E\rightarrow F$, given any contractive $\phi:P\rightarrow F$, we can lift this to a contractive $\varphi:P\rightarrow E$ with $\psi\varphi = \phi$. (If someone can make a nice commutative diagram, go ahead and edit this!)


**Claim:** The scalar field (say $\mathbb C$, but also works for $\mathbb R$) is not projective.


**Proof:** Let $f:c\_0\rightarrow\mathbb C$ be the contractive functional $f(x) = \sum\_{n=1}^\infty 2^{-n} x\_n$ for $x=(x\_n)\in c\_0$. This induces an isometric isomorphism $c\_0 / \ker(f) \cong \mathbb C$. So if $\mathbb C$ is projective, we can find a *contractive* $g:\mathbb C\rightarrow c\_0$ with $fg=1$. That is, $g=(g\_n)\in c\_0$ is a norm-one vector with $f(g) = \sum 2^{-n} g\_n =1$, which is impossible.



> 
> **Question:** Are there any projective Banach spaces?
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It seems to me that the problem is insisting upon *contractive* morphisms. In the proof above, if we just need, for each $\epsilon>0$, to find $g$ with $\|g\|<1+\epsilon$, then this is no problem. Is anything known in this generalised setting? (It's easy to see that then $\ell\_1(\Gamma)$ is always projective, if I allow myself this wiggle-room).



Projective Banach spaces
You essentially answered your first question yourself: the ground field is a (contractive) retract of any nonzero Banach space by Hahn-Banach. If there were a non-zero projective Banach space in your sense then the ground field would be projective as well.


On the other hand: It is a theorem due to Köthe and Pełczyński that every projective Banach space (lifting over surjective maps in the additive category of Banach spaces) is isomorphic to $\ell^1{(S)}$ for some $S$. I don't know how well the norm of the isomorphism is controlled, but as all these spaces already satisfy your relaxed condition, you won't find any others.


The references for the second paragraph are:


* A. Pełczyński, *Projections in certain Banach spaces*, Studia Math. **19** (1960), 209–228. MR0126145.
* G. Köthe, *Hebbare lokalkonvexe Räume*, Math. Ann. **165** (1966), 181–195. MR0196464.


