Gram's Inequality

Let , ..., be real integrable functions over the closed interval , then the determinant of their integrals satisfies

|int_a^bf_1^2(x)dx int_a^bf_1(x)f_2(x)dx ... int_a^bf_1(x)f_n(x)dx; int_a^bf_2(x)f_1(x)dx int_a^bf_2^2(x)dx ... int_a^bf_2(x)f_n(x)dx; | | ... |; int_a^bf_n(x)f_1(x)dx int_a^bf_n(x)f_2(x)dx ... int_a^bf_n(x)f_n(x)dx|>=0.

See also

Gram-Schmidt Orthonormalization

Explore with Wolfram|Alpha

More things to try:

References

Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, p. 1100, 2000.

Referenced on Wolfram|Alpha

Gram's Inequality

Cite this as:

Weisstein, Eric W. "Gram's Inequality." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/GramsInequality.html

Subject classifications