## Schwarz's Inequality

 (1)

Written out explicitly
 (2)

with equality Iff with a constant. To derive, let be a Complex function and a Complex constant such that for some and . Since ,

 (3)

with equality when . Set
 (4)

so that
 (5)

Plugging (5) and (4) into (3) then gives
 (6)

 (7)

 (8)

so
 (9)

Bessel's Inequality can be derived from this.

References

Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 11, 1972.

Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 527-529, 1985.