Let *U*_{1} and *U*_{2} be two subspaces of the same vector space *V*. The **sum** of these subspaces, denoted *U*_{1} + *U*_{2}, is the set of all the sums
,
where
and
.
If
,
the sum is a **direct sum** and is denoted
.
If
,
then *U*_{1} and *U*_{2} are **supplementary** subspaces.

- 1.
- Take and . We have: .
- 2.
- Let be the vector space of all the functions defined on . Denote by (resp. ) the subspace of all the odd (resp. even) functions from to itself. Then .

This is linked to Thm 4.5.