TITLE: Must an injective or surjective map in an infinite dimensional vector space be a bijection?
QUESTION [12 upvotes]: If we have some finite dimensional vector space V and linear transformation $F: V\to V$, and we know that F is injective, we immediately know that it is also bijective (same goes if we know that F is surjective).
I'm curious if a same rule applies if V is infinite dimensional vector space, and we know that F is injective/surjective, does it again immediately imply that F is also bijective (intuitively I think it does)?

REPLY [4 votes]: Abstract example: take a basis $(b_i)_{i\in I}$ of the space. Let be $J\subset I$ with $|J| = |I|$ and $\sigma:I\longrightarrow J$ a bijection. The linear function defined by
$$S(b_i) = b_{\sigma(i)},$$
is injective but not surjective while
$$T(b_i) = b_{\sigma^{-1}(i)},i\in J,$$
$$T(b_i) = 0,i\in I\setminus J,$$
is surjective but not injective.