Characterization of direct sums

Theorem. Let $M$ and $N$ be linear subspaces of a vector space $Y$ . Then

Y=M\oplus N

if and only if every $y\in Y$ admits a unique representation

y=a+b \quad\text{with } a\in M,\ b\in N.

Remarks

Context. This result explains why direct sums behave like "coordinate decompositions" with respect to the two subspaces.

Proof sketch.

( $\Rightarrow$ ) If $y=a+b=a'+b'$ , then $a-a'=b'-b\in M\cap N=\{0\}$ , so $a=a'$ and $b=b'$ .
( $\Leftarrow$ ) Existence gives $Y=M+N$ , since $M,N\subseteq Y$ . Uniqueness gives $M\cap N=\{0\}$ : if $x\in M\cap N$ , then $x=x+0=0+x$ , so $x=0$ .