Proof of Cauchy–Schwartz Inequality

The Cauchy-Schwartz inequality is one of the important geometric properties of Hilbert space. The proof of this inequality is as follows:

First, let's examine the left-hand side of the inequality.

Let us now consider the right-hand side of the inequality. We can write the unit operator, which is the sum of all orthonormal basis kets in the Hilbert space.

Let's look at an example that satisfies this inequality.

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