- Published on
Operators on Infinite Dimensional Vector Spaces 6
- Authors
- Name
- Malachy Reynolds
- @MalachyReynolds
Projections
Definition 2.12: An operator is called a projection if it is idempotent, i.e. .
Definition 2.13: A self-adjoint projection i.e. is called an orthogonal projection.
Lemma 2.14: Let be a projection. Then is a projection, and:
In particular, is closed.
Proof:
First we see that is a projection. Indeed:
Then we have that:
and from a similar computation follows. Since , we have found that . On the other hand for any we have that , i.e. , so that . Thus , and . The equality follows from reversing the roles of and and observing that .
Corollary 2.15: There is a one-to-one correspondence between orthogonal projections and closed subspaces, associating subspaces with the projection on and a projection with the subspace . We have for every subspace :
Proof: Lemma 2.14 is one side of the implication, associating a subspace with the closed subspace. For the other side, the projection theorem tells us that for closed subspaces , so that any can be written uniquely as , and . The map is an orthogonal projection with (verify orthogonality), and we can apply Lemma 2.14 to it.