Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 318 / 5 / 1 / b

Table of contents
- Solution b

Solution

The defining relation for

P_{U}

gives

(f - P_{U} f, v) = 0 \forall v \in U_{n} .

Part (a) therefore shows that

P_{U} f

is the best approximation. Moreover,

P_{U} f

and

f - P_{U} f

are orthogonal, so

∥ f ∥^{2} = ∥ P_{U} f ∥^{2} + ∥ f - P_{U} f ∥^{2} .

Thus orthogonal projection is a contraction:

∥ P_{U} f ∥ \leq ∥ f ∥ .

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