Let v 1 , … , v k {\displaystyle {}v_{1},\ldots ,v_{k}} denote a basis of U {\displaystyle {}U} . We can extend this basis, according to fact, to a basis v 1 , … , v k , v k + 1 , … , v n {\displaystyle {}v_{1},\ldots ,v_{k},v_{k+1},\ldots ,v_{n}} of V {\displaystyle {}V} . Then
fulfills all the properties of a direct sum.
denote a
basis
of 
. We can extend this basis, according to
fact,
to a basis 
of 
. Then
