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Problem 30. Let H be a Hilbert space and M be a closed subspace of H. Denoting by P: H – M the orthogonal projection of H onto M, prove that, for any x, y E H, (Px, y) = (x, Py). (This is telling us that P is self-adjoint).

Problem 30. Let H be a Hilbert space and M be a closed subspace of H. Denoting by P: H – M the orthogonal projection of H onto M, prove that, for any x, y E H, (Px, y) = (x, Py). (This is telling us that P is self-adjoint).

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter6: Vector Spaces
Section: Chapter Questions
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Problem 30. Let H be a Hilbert space and M be a closed subspace of H. Denoting by P : H → M the orthogonal projection of H onto M, prove that, for any x, y E H, (Px, y) = (x, Py). (This is telling us that P is self-adjoint).
Transcribed Image Text:Problem 30. Let H be a Hilbert space and M be a closed subspace of H. Denoting by P : H → M the orthogonal projection of H onto M, prove that, for any x, y E H, (Px, y) = (x, Py). (This is telling us that P is self-adjoint).
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