Projection
Definition 1. Let such that , and let where is a vector space. Then is the projection on if, whenever with , we have .
Theorem 1
Theorem 1. Let such that where is a vector space. If is the projection on , then and , so .
Proof. Let . Then clearly . Let . Then for some . If we denote for , then . Thus .
Let , and let denote for . Then . Thus . Let . Then clearly , so . Thus .
Theorem 2
Theorem 2. Let where is a vector space. Then is a projection .
Proof. Let such that .
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Suppose that is the projection on . Let denote for . Then . Thus .
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We claim that where . Note that and note that . Then . Thus and .
For any , we have and . Thus , so . This means that .
Let denote for some and . Then and . Thus , so is the projection on .