Definition:Complementary Subspace

Definition

Let $X$ be a vector space.

Let $N, Y \subseteq X$ be subspaces.

Then $Y$ is a complementary subspace of $N$ if and only if $X$ is the direct sum:

$X = N \oplus Y$

That is, if and only if for each $x \in X$, there exist unique $n \in N$ and $y \in Y$ such that:

$x = n + y$

Also see

• Existence of Complementary Subspace

Sources

• 2002: Peter D. Lax: Functional Analysis: $2.2$: Index of a Linear Map