orthogonal decomposition theorem
is closed :
Since is complete (http://planetmath.org/Complete) and is closed, is a subspace of . Therefore, for every , there exists a best approximation of in , which we denote by , that satisfies (see this entry (http://planetmath.org/BestApproximationInInnerProductSpaces)).
This allows one to write as a sum of elements in and
which proves that
Moreover, it is easy to see that
since if then , which means .
We conclude that .
|Title||orthogonal decomposition theorem|
|Date of creation||2013-03-22 17:32:34|
|Last modified on||2013-03-22 17:32:34|
|Last modified by||asteroid (17536)|
|Synonym||closed subspaces of Hilbert spaces are complemented|