 Solution

ID: past-exam-of-the-algebra-and-analysis-ii-for-mathematics-and-philosophy-course-of-the-university-of-oxford/2013/2/b/i/solution

R^{n}

, with

a = (a_{1}, \dots, a_{n})

and

b = (b_{1}, \dots, b_{n})

. If

b = 0

, both sides of the desired inequality are zero. Otherwise take the single unit vector

v = b /∥ b ∥

in part (a). Then

∥ a ∥^{2} \geq \frac{⟨ a , b ⟩ ^{2}}{∥ b ∥ ^{2}},

(i = 1 \sum n a_{i}^{2}) (i = 1 \sum n b_{i}^{2}) \geq (i = 1 \sum n a_{i} b_{i})^{2} .

This is the real Cauchy-Schwarz inequality. Equality holds when the two vectors are linearly dependent.

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