Past exam of the algebra, analysis and differential equations II course of the University of Oxford / 2005 / Algebra / 2 / i

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- Solution i

Solution

A real inner product is a symmetric bilinear map

⟨ \cdot, \cdot ⟩ : V \times V \to R

with

⟨ v, v ⟩ > 0

for

v \neq = 0

∥ v ∥ = ⟨ v, v ⟩

W = span {e_{1}, \dots, e_{k}}

and

P v = \sum_{i} ⟨ v, e_{i} ⟩ e_{i}

. Orthonormality gives

v - P v ⊥ W

, hence the Pythagorean identity gives

∥ v ∥^{2} = ∥ v - P v ∥^{2} + \sum_{i} ⟨ v, e_{i} ⟩^{2} .

This proves Bessel's inequality. Equality holds exactly when $v \in W$ .

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