Diversions in Mathematics 3b: The Cauchy-Schwarz inequality (part 2)

In the previous post, we introduced the Cauchy-Schwarz inequality purely as a statement about real numbers:

\displaystyle \sum_{k=1}^n x_k y_k \le \left( \displaystyle \sum_{k=1}^n x_k^2 \right)^{1/2} \left( \displaystyle \sum_{k=1}^n y_k^2 \right)^{1/2}.

We proved this inequality using mathematical induction. However, one could argue that the method did not yield much insight. As our proof was entirely algebraic, it is not easy to see why the sums that appear in the inequality are reasonable quantities to consider, and there is no hint as to how they could arise naturally in mathematical problems. Therefore, in this post, we will look at the Cauchy-Schwarz inequality from a geometric perspective, which reveals a much richer structure. Moreover, we will see how the inequality arises naturally from a basic optimisation problem.

Note: I continue with the convention from the previous post that a quantity x is called positive if x \ge 0. We say that x is strictly positive if x > 0.

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