Cauchy-Schwarz Inequality for Integrals
Cauchy-Schwarz Inequality for Integrals
The Cauchy–Schwarz inequality for integrals states that dxdx≳ 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 f(x)dxg(x)x⩾(x)g(x)dx. The limiting case of equality is obtained when and are linearly dependent, for instance when ).
b
∫
a
2
f(x)
b
∫
a
2
g(x)
2
f(x)g(x)dx
b
∫
a
[a,b]
AB≥
2
|
2
|
2
(A.B)
<ff><gg>⩾<fg
2
>
b
∫
a
2
|
b
∫
a
2
|
b
∫
a
*
f
2
|
f(x)
g(x)
g(x)=constantf(x
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 .
[-1,1]
f(x)
g(x)
[-5,5]