 Solution

ID: past-exam-of-the-algebra-analysis-and-differential-equations-2-course-of-the-university-of-oxford/2009/algebra/2/a/solution

For any

w \in W

, the decomposition

v - w = T v + (P v - w)

∥ v - w ∥^{2} = ∥ T v ∥^{2} + ∥ P v - w ∥^{2} \geq ∥ T v ∥^{2} .

Equality is attained at

w = P v

. Thus

w \in W in f ∥ v - w ∥ = ∥ T v ∥

, and the minimizer in

W

is uniquely

P v

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