From u ∈ U ∩ U ⊥ {\displaystyle {}u\in U\cap U^{\perp }} we get directly
thus u = 0 {\displaystyle {}u=0} . This means that the sum direct. Let u 1 , … , u k {\displaystyle {}u_{1},\ldots ,u_{k}} be an orthonormal basis of U {\displaystyle {}U} . We extend it to an orthonormal basis u 1 , … , u n {\displaystyle {}u_{1},\ldots ,u_{n}} of V {\displaystyle {}V} . Then we have
Therefore, V {\displaystyle {}V} is the sum of the linear subspaces.
we get directly
.
This means that the sum direct. Let 
be an
orthonormal basis
of 
. We extend it to an orthonormal basis 
of 
. Then we have
is the sum of the linear subspaces.
