Existence of Complementary Subspace

Theorem
Let $X$ be a vector space.

Let $N \subseteq X$ be a subspace.

Then $N$ has a complementary subspace.

That is, there exists a subspace $Y \subseteq X$ such that:
 * $X = N \oplus Y$

Proof
Let $S$ be the set of all subspaces $V \subseteq X$ such that:
 * $N \cap V = \set 0$

By Zorn's Lemma, $\struct {S, \subseteq}$ has a maximal element $Y$.

We claim:
 * $X = N \oplus Y$

Indeed, if there would be an $x \in X \setminus \paren {N \oplus Y}$, then:
 * $Y \subsetneq \map \span {Y \cup \set x} \in S$

where $\span$ denotes the linear span.

This contradicts the maximality of $Y$.