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Cauchy-Schwarz Inequality for Integrals

The Cauchy–Schwarz inequality for integrals states that for two real integrable functions in an interval . This is an analog of the vector relationship , which is, in fact, highly suggestive of the inequality expressed in Hilbert space vector notation: . For complex functions, the Cauchy–Schwarz inequality can be generalized to . The limiting case of equality is obtained when and are linearly dependent, for instance when ).
This Demonstration shows examples of the Cauchy–Schwarz inequality in the interval , in which and are polynomials of degree four with coefficients in the range .

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Snapshot 1: inequality of an order of magnitude
Snapshot 2: limiting case of equality since and are proportional
Snapshot 3: case of two orthogonal functions
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