Characterization of direct sums

Theorem. Let $M$ and $N$ be linear subspaces of a vector space $X$, and let $Y\subset X$. Then

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

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

Proof sketch.

• ($\Rightarrow$) If $y=a+b=a'+b'$ with $a,a'\in M$ and $b,b'\in N$, then $a-a'=b'-b\in M\cap N=\{0\}$, so $a=a'$ and $b=b'$.
• ($\Leftarrow$) If every $y$ has a unique decomposition, then certainly $Y\subset M+N$ and $M+N\subset Y$ (by definition of $Y$), and uniqueness forces $M\cap N=\{0\}$ since $x=x+0=0+x$ implies $x=0$ for $x\in M\cap N$.