Past exam of the mathematics course of the University of Cambridge / 2023 / ib / Paper 4 / 6B / b

Table of contents
- Solution b

Solution

Part (a) shows that

f - p_{n}^{*}

is orthogonal to

p_{n}^{*} \in P_{n}

. Since

f = (f - p_{n}^{*}) + p_{n}^{*},

the Pythagorean theorem in an inner-product space immediately gives

∥ f ∥^{2} = ∥ f - p_{n}^{*} ∥^{2} + ∥ p_{n}^{*} ∥^{2} .

This is the norm decomposition associated with orthogonal projection onto a finite-dimensional subspace.

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