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 .