Introduction Notes

# Consider for t r f t y tw x2 y x tw2 then by

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: and x ∈ Cm , then a vector y ∈ V0 satisﬁes y − x ⊥ V0 if and only if ￿y − x￿ ≤ inf ￿v − x￿ . v ∈V 0 Moreover, there is only one such point y in V0 with these properties. Proof. First we assume v ∈ V0 , y − x ⊥ x. Choose any u ∈ V0 , u ￿= y , then u = y + w with w ￿= 0. Consider for t ∈ R f (t) = ￿y + tw − x￿2 = ￿(y − x) + tw￿2 then by the orthogonality of y − x and w ∈ V0 , f (t) = ￿y − x￿2 + t2 ￿w￿2 which has a global minimum at t = 0. Since w ￿= 0, f (0) = ￿P x − x￿2 < f (1) = ￿u − x + w￿2 = ￿u − x￿2 , so among all u ∈ V0 the unique minimizer for the distance ￿u − x￿ is y . Conversely, assume that there is y ∈ V0 which satisﬁes that ￿y − x￿ ≤ inf v∈V0 ￿v − x￿. Choosing any w ∈ V0 \ {0} and α ∈ C then gives that f (t) = ￿y + tαw − x￿2 is minimal at t = 0, so 0 = f ￿ (0) = 2Reα￿w, y − x￿. Taking the supremum of the real part over all complex α gives that ￿w, y − x￿ = 0 which me...
View Full Document

## This note was uploaded on 01/16/2014 for the course MATH 6304 taught by Professor Bernhardbodmann during the Fall '12 term at University of Houston.

Ask a homework question - tutors are online