A question about vectorial spaces of infinite dimension.

Question

Let $V$ be a $k$-vectorial space of finite dimension $n$, every subspace $U \subseteq V$, can be seen as a direct summand of $V$, it is sufficient choose a basis of $U$ and by completing to a basis of $V$, the elements that don't belong to the basis of $U$, generate a subspace $U'$ such that $U \oplus U'=V$. My question is: is this also true when $V$ is an infinite dimensional vectorial space?

Answer 1

The existence of a basis for any infinite-dimensional vector spaces is equivalent to Zorn's lemma and the Axiom of Choice. So "yes", if you accept AC, and "it depends", if you don't.