Direct sum ($\oplus$)

The direct sum $U\oplus W$ of two vector spaces consists of pairs $(u,w)$ with componentwise operations. A vector space is an internal direct sum of subspaces $U$ and $W$ exactly when every vector has a unique expression $u+w$ with $u\in U$ and $w\in W$.

An orthogonal direct sum is a direct sum whose distinct summands are mutually orthogonal. For $v\in V$ and $w\in W$, the Pythagorean identity gives $\Vert v+w{\Vert }^2=\Vert v{\Vert }^2+\Vert w{\Vert }^2$. In a Hilbert space, a closed subspace of a Hilbert space and its orthogonal complement give an orthogonal direct sum of the entire space. The component maps are orthogonal projections and have operator norm one when their ranges are nonzero.

A submodule $N$ of a module $M$ is a direct summand if some submodule $L$ gives $M=N\oplus L$. Equivalently, the inclusion has a left inverse R-module homomorphism, or $N$ is the image of an idempotent endomorphism of $M$.

For finite-dimensional subspaces $U,W$ of a common vector space,

For every subspace $U$ of a finite-dimensional vector space $V$, extending a basis of $U$ to a basis of $V$ produces a subspace $W$ such that $V=U\oplus W$.