m10435 Projection Theorem 2.12 2002/01/07 2009/05/31 18:10:57.254 GMT-5 Phil Schniter Phil Schniter schniter@ee.eng.ohio-state.edu Phil Schniter Phil Schniter schniter@ee.eng.ohio-state.edu Charlet Reedstrom Charlet Reedstrom charlet@rice.edu Phil Schniter Phil Schniter schniter@ee.eng.ohio-state.edu Hilbert space orthogonality principle orthogonal projection Science and Technology This module introduces projection theorem. en The orthogonal projection of y onto S, where S is a closed subspace of V, is y ̂ i i x i y x i s.t. x i is an ON basis for S. Orthogonal projection yields the best approximation of y in S: y ̂ argmin x S y x The approximation error e y y ̂ obeys to orthogonality principle: ⊥ e S We illustrate this concept using V 3 below but stress that the same geometrical interpretation applies to any Hilbert space.

A proof of the orthogonality principle is: ⇔ ⊥ e S i e x i 0 y y ̂ x i 0 y x i y ̂ x i y x i j j x j y x j x i y x i j j x j y x j x i y x i j j y x j d i − j y x i y x i