 Solution

ID: past-exam-of-the-algebra-analysis-and-differential-equations-2-course-of-the-university-of-oxford/2005/algebra/2/i/solution

⟨ \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}

v - P v ⊥ W

∥ 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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