In Linear Algebra Done Right by Sheldon Axler, Section 3.B Question 20 says

> 
> Suppose $W$ is finite-dimensional and $T \in \mathcal{L}(V, W)$. Prove that $T$ is injective iff there exists $S \in \mathcal{L}(W, V)$ such that $ST$ is the identity map on $V$.
> 
> 
> 

It doesn't specify $V$ as being finite-dimensional, and maybe it's not needed for the proof, but I was wondering if it's possible for $V$ to be infinite-dimensional? Also in that section, there is a proposition stating:

> 
> Suppose V and W are finite-dimensional vector spaces such that dim V > dim W. Then no linear map from V to W is injective.
> 
> 
> 

This is specified for finite-dimensional spaces $V$ and $W$. It seems that for infinite-dimensional $V$ this would then definitely be the case, but so why isn't that stated? I was wondering if this is true but maybe proved later in the book or elsewhere, or if it's not true at all.