 Solution

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

R^{2 n}

, take the two unit vectors

v_{even} = \frac{1}{n} (0, 1, 0, 1, \dots, 0, 1), v_{odd} = \frac{1}{n} (1, 0, 1, 0, \dots, 1, 0) .

Their supports are disjoint, so they are orthonormal. Apply part (a) to

a = (a_{1}, \dots, a_{2 n})

\sum_{i = 1}^{2 n} a_{i}^{2} \geq \frac{1}{n} (\sum_{i = 1}^{n} a_{2 i})^{2} + \frac{1}{n} (\sum_{i = 1}^{n} a_{2 i - 1})^{2} .

Multiplying by

n

gives

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

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